CELL SIZE HOMEOSTASIS AND OPTIMAL VIRAL STRATEGIES

FOR HOST EXPLOITATION

by

Cesar Augusto Vargas-Garcia

A dissertation submitted to the Faculty of the University of Delaware in partialfulfillment of the requirements for the degree of Doctor of Philosophy in Electrical andComputer Engineering

Fall 2017

c© 2017 Cesar Augusto Vargas-GarciaAll Rights Reserved

ACKNOWLEDGEMENTS

I want to thank my advisor Abhyudai Singh. He discovered my professional

potential and helped me to achieve this important goal in my life. I am grateful to

the Dean, the Faculty, and the Staff of the Department of Electrical and Computer

Engineering for providing their assistance and support through the years of my Ph.D.

program.

I want to thank also my co-advisor and friend, Dr. Ryan Zurakowski. He gave

me the opportunity to start and enjoy this field. Also he encouraged me to be resilient

in pursuing my degree in the hard days.

I want to thank to professor and close friend Henry Arguello for all his support

and advice through this years.

I also want to thank my wife and daughter, Neyla Johanna and Victoria for

their support and encouragement during my studies. They provided me the home to

rest after every hard day.

My students and alma-mater group HDSP gave me the motivation and encour-

agement to make the best effort in my research. They have been my friends and part

of my family during this time. I appreciate their collaboration and company in this

part of my life.

Special thanks to my friends and colleagues Mohammad Soltani and Khem

Ghusinga for their support, friendship and collaborations in uncountable and exciting

projects which are nowadays the core of my research.

iii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Chapter

1 PART 1: CELL SIZE CONTROL AND GENE EXPRESSIONHOMEOSTASIS IN SINGLE-CELLS . . . . . . . . . . . . . . . . . . 1

1.1 Cell-size regulation: going beyond phenomenological models . . . . . 11.2 Why do organisms control size? . . . . . . . . . . . . . . . . . . . . . 41.3 Living with size variations: gene expression homeostasis . . . . . . . . 5

2 CONDITIONS FOR CELL SIZE HOMEOSTASIS: ASTOCHASTIC HYBRID SYSTEMS APPROACH . . . . . . . . . 13

2.1 Timer-dependent growth and division . . . . . . . . . . . . . . . . . . 142.2 Size-dependent growth rate . . . . . . . . . . . . . . . . . . . . . . . 162.3 Size-dependent division rate . . . . . . . . . . . . . . . . . . . . . . . 19

3 A MECHANISTIC STOCHASTIC FRAMEWORK FORREGULATING BACTERIAL CELL DIVISION . . . . . . . . . . . 25

3.1 Distribution of the cell-division time given newborn cell size . . . . . 293.2 Distribution of the volume added between divisions . . . . . . . . . . 313.3 Higher order moments of added volume . . . . . . . . . . . . . . . . 34

4 PART 2: OPTIMALITY IN HOST-VIRUS SYSTEMS . . . . . . 37

5 OPTIMAL ADSORPTION RATE: IMPLICATIONS OF THESHIELDING EFFECT . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 Traditional virus dynamics model . . . . . . . . . . . . . . . . . . . . 415.2 Modeling the shielding effect . . . . . . . . . . . . . . . . . . . . . . . 435.3 Single virus dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Competition dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iv

6 THE EFFECT OF MULTIPLICITY OF INFECTION ON THETEMPERATENESS OF A BACTERIOPHAGE: IMPLICATIONSFOR VIRAL FITNESS . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Why do Bacteriophages display Temperateness? . . . . . . . . . . . . 546.3 Probability of survival of a lysogen . . . . . . . . . . . . . . . . . . . 576.4 The Effect of Multiplicity of infection (MOI) . . . . . . . . . . . . . . 58

7 CONDITIONS FOR INVASION OF SYNAPSE-FORMING HIVVARIANTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.1 HIV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.2 Modeling Synaptic Virus . . . . . . . . . . . . . . . . . . . . . . . . . 667.3 Competition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8 OPTIMAL MULTI-DRUG APPROACHES FOR REDUCTIONOF THE LATENT POOL IN HIV . . . . . . . . . . . . . . . . . . . 75

8.1 HIV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Optimal Control and Simulations . . . . . . . . . . . . . . . . . . . . 83

9 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

v

ABSTRACT

The first part of this thesis address a question formulated more than 80 years

ago (and still remains elusive): how does a cell control its size? Growth of a cell and its

subsequent division into daughters is a fundamental aspect of all cellular living systems.

During these processes, how do individual cells correct size aberrations so that they

do not grow abnormally large or small? How do cells ensure that the concentration of

essential gene products are maintained at desired levels, in spite of dynamic/stochastic

changes in cell size during growth and division?

In chapter 1, we introduce the reader to the field of cell size/content homeostasis.

We review how advances in singe-cell technologies and measurements are providing

unique insights into these questions across organisms from prokaryotes to human cells.

More specifically, how diverse strategies based on timing of cell-cycle events, regulating

growth, and number of daughters are employed to maintain cell size homeostasis. We

further discuss how size-dependent expression or gene-replication timing can buffer

concentration of a gene product from cell-to-cell size variations within a population.

In chapter 2, we propose the use of stochastic hybrid systems as a framework for

studying cell size homeostasis. We assume that cell grows exponentially in size (volume)

over time and probabilistic division events are triggered at discrete time intervals. We

first consider a scenario, where a timer (i.e., cell-cycle clock) that measures the time

since the last division event regulates both the cellular growth and division rates. We

also study size-dependent growth / division rate regulation mechanisms. We provide

bounds on different statistical indicators (mean, variance, skewness, etc). Additionally,

we assess the effect of different physiological parameters (growth rate, partition errors,

etc) on cell size distribution.

vi

Chapter 3 introduces a mechanistic model that might explain the recently un-

covered added principle, i.e., selected species add a fixed size (volume) from birth to

division, irrespective of their size at birth. To explain this principle, we consider a

timekeeper protein that begins to get stochastically expressed after cell birth at a rate

proportional to the volume. Cell-division time is formulated as the first-passage time

for protein copy numbers to hit a fixed threshold. Consistent with data, the model

predicts that the noise in division timing increases with size at birth. We show that the

distribution of the volume added between successive cell-division events is independent

of the newborn cell size. This fact is corroborated through experimental data avail-

able. The model also suggest that the distribution of the added volume when scaled

by its mean become invariant of the growth rate, a fact also verified through available

experimental data.

In part 2 of this thesis, we study which strategies are implemented by a viral

species, ranging from bacteriophages to human immunodeficiency virus (HIV), in order

to exploit host resources. In chapter 4, we review the classical theory of viral-host

dynamics and describe the key knobs that viruses tweak to exploit a cell population.

This theory suggest that viruses might evolved to have infinite infectivity and virulence.

In the case of infectivity, chapter 5 gives an alternative to infinite infectivity: virus will

evolve to moderate infectivity because of local interactions. As an example, we study

a phage attacking a bacterial population. We include the effect of local interactions by

assuming that the phage needs to scape from bacterial death remains (debris).

Infinite virulence is also challenged as evolutionary alternative for viral propa-

gation. In chapter 5 we study environments where availability of susceptible bacteria

fluctuates across time. Under such scenarios bacteria behaves contrary to classical

ecology theory: phages evolve to a moderate virulence (lysis time). We present this

insights through the use of the stochastic hybrid system framework.

In chapter 7, we present a mathematical model of HIV transmission including

cell-free and cell-cell transmission pathways. A variation of this model is considered

including two populations of virus. The first infects cells only by the cell-free virus

vii

pathway, and the second infects cells by either the cell-free or the cell-cell pathway

(synapse-forming virus). Synapse-forming HIV is shown to provide an evolutionary

advantage relative to non synapse-forming virus when the average number of virus

transmitted across a synapse is a sufficiently small fraction of the burst size.

HIV disease is well-controlled by the use of combination antiviral therapy (cART),

but lifelong adherence to the prescribed drug regimens is necessary to prevent viral re-

bound and treatment failure. Populations of quiescently infected cells form a “latent

pool” which causes rapid recurrence of viremia whenever antiviral treatment is inter-

rupted. A “cure” for HIV will require a method by which this latent pool may be

eradicated. Current efforts are focused on the development of drugs that force the

quiescent cells to become active. Previous research has shown that cell-fate decisions

leading to latency are heavily influenced by the concentration of the viral protein Tat.

While Tat does not cause quiescent cells to become active, in high concentrations it

prevents a newly infected cell from becoming quiescent. In chapter 8, we introduce a

model of the effects of two drugs on the latent pool in a patient on background sup-

pressive therapy. The first drug is a quiescent pool stimulator, which acts by causing

quiescent cells to become active. The second is a Tat analog, which acts by preventing

the creation of new quiescently infected cells. We apply optimal control techniques to

explore which combination therapies are optimal for different parameter values of the

model.

viii

Chapter 1

PART 1: CELL SIZE CONTROL AND GENE EXPRESSIONHOMEOSTASIS IN SINGLE-CELLS

1.1 Cell-size regulation: going beyond phenomenological models

Size plays an important role in cellular processes and functions of a cell [1], and

therefore should be actively maintained. Indeed, cell size distribution of proliferating

cells is known to be stable through generations, suggesting regulation of growth and

division to correct deviations from a desired cell size. Earlier attempts towards under-

standing cell size control was based on population-averaged data on model organisms,

bacteria and yeast, and led to proposition of phenomenological models of cell size con-

trol. In particular, three models were hypothesized: Timer – a constant time between

successive divisions, Sizer – cell division upon attainment of a critical size, and Adder

– a constant size addition between consecutive generations. However, validation of

these in various organisms remained inconclusive. With recent advances in single-cell

technologies, high throughput measurements of cell size over several cell-cycles of in-

dividual cells can be made, generating correlation data between different parameters

such as cell size at birth, growth rate, cell size at division, division time, etc. This data

has stimulated reexamination of phenomenological models of cell size homeostasis.

For a broad range of microbes, an individual cell grows exponentially over time

with a constant growth rate per size [2, 3]. A Timer based mechanism to control

division can thus be precluded since it would not be homeostatic [4,5]. Consistent with

it, analysis of data for several bacterial species reveals a negative correlation between

division time and cell size at birth, implying presence of a size control during cell-cycle.

Further investigation reveals the phenomenological strategy: size added from birth to

division is uncorrelated with cell size at birth, which is inconsistent with a Sizer model

1

and validates an Adder model (Fig. 1.1a) [6–12]. Existence of such phenomenological

models, however, only provides limited perspective since it does not specify how other

landmark cell-cycle events (e.g., initiation of DNA replication, assembly of division

apparatus) are coordinated with division, and whether the size control is applied from

birth to division or between two other cell-cycle events.

Studies on E. coli have proposed several formulations that couple initiation of

DNA replication to division while being consistent with an Adder between birth and

division. One model postulates that size control is primarily exerted over the timing

of initiation of DNA replication such that a constant volume per origin of replication

is added between two consecutive initiation events. The corresponding division is

assumed to occur a fixed time (C+D period; C–time to replicate DNA, D–time between

end of replication to division) after initiation [13–15]. Another proposition, which

suggests that initiation of DNA replication occurs at a constant size per origin and C+D

period depends upon the growth rate, shows that the Adder model is valid only for fast

growth rates and the size control behaves as a Sizer for slow growth conditions [16]. A

third model argues that for slow growing cells, size control is exerted at two sub-periods

(the time from birth to initiation, and the D period) whereas the C period resembles

a Timer [17]. So far none of these models have been conclusively validated or falsified,

and it would be worthwhile to carry out experiments to this end.

Similar couplings between important cell-cycle events and division have been

explored in other organisms as well. For C. crescentus, pre-constriction and post-

constriction periods have been examined for size control, showing that it obeys a

mixer model wherein the time until constriction acts as a Timer followed by the post-

constriction period regulated via an Adder [18]. Likewise, the cell-cycle of budding

yeast has been investigated by dividing it in two distinct periods: time until G1/S

transition, and time from the G1/S transition to division. Despite having an overall

Adder between birth to division [19, 20], independent control of both these periods is

proposed. In particular, the first period is shown to be dependent upon the size of

2

mother cell, and the second period is controlled by the size of the daughter [20]. Col-

lectively, dissecting the cell-cycle in biologically relevant periods for various organisms

provides key insights in to how division might be coordinated with other events.

An important step moving forward is to understand molecular mechanisms that

implement size control over timing of cell-cycle events. To this end, two generic themes

have been proposed. One approach is to accumulate a protein in size dependent man-

ner up to a threshold. Some notable examples of this include FtsZ to control Z-ring

formation, [21, 22], DnaA to control timing of initiation in E. coli [13], and Cdc25

to regulate timing of mitotic entry in S. pombe [23]. Another way to implement a

size control over timing is to dilute a protein until a critical level as cell grows in

size. A prominent example of this strategy is Whi5 for control of G1 duration in S.

cerevisiae [19, 20, 24, 25]. Interestingly, an alternative model shows that an Adder-like

behavior can also arise from a very different mechanism of maintaining a constant sur-

face area to volume ratio [26]. Apparently, the nutrient intake imposes constraints on

this ratio by affecting the synthesis of surface material. The candidate molecules that

carry out such function have not been identified yet. It is plausible that molecular

players underlying important cell-cycle events interact with each other, and therefore

an overarching framework may emerge with further research.

How is size control implemented in multicellular organisms? Arguably, these

organisms operate in a more complex environment than bacteria and budding yeast;

thus, size control strategies adopted by their cells are expected to be affected by physical

constraints and thereby be relatively more complicated. Recent data indeed suggests

that mammalian cells have different size control strategy in the G1 duration than

budding yeast. This strategy phenomenologically resembles a Sizer for small cells, but

Adder for larger cells [27]. Examining the data further reveals that for mammalian cells,

not only the time spent in G1, but also the growth rate are negatively correlated with

size at birth [27] (Fig. 1.1b). This observation has been strengthened by recent work

showing size-dependent regulation of growth rate [30,31]. The molecular underpinnings

of growth rate control are not well understood, although access to nutrients and physical

3

constraints are expected to play an important role.

In a stark contrast to size control on timing of cell-cycle events and growth rate,

the green alga C. reinhardtii controls the number of daughter cells it produces upon

division in a size controlled manner. It has a prolonged G1 phase in which the size of

a newborn cell increases by several folds. The cell then undergoes multiple divisions

producing 2n daughter cells. The number of divisions n, on average, is large for large

mother cells, so that the daughter cells are close to a target size (Fig. 1.1c) [28, 29].

The molecular mechanism for regulating the number of divisions in C. reinhardtii is

suggested to rely on concentration dilution. Here, a cyclin dependent kinase CDKG1

is produced towards the end of G1 phase and its production depends on the size of the

cell. With each division, this protein is degraded and further divisions stop once its

concentration goes below a threshold [28].

1.2 Why do organisms control size?

Recent experiments in proliferating animal cells unravel an interesting connec-

tion between cell size homeostasis and cellular fitness. By sorting a population of

growing cells based on their sizes, it was shown that cellular proliferation (i.e., fitness)

is low for small and large cells, but high at intermediate sizes [32, 33] (Fig. 1.2). The

decrease in the fitness for large cells could not be attributed to higher cell-death since

the apoptosis rates here were similar for large and average–sized cells. Notably, a sim-

ilar optimality also exists for mitochondrial activity (metabolism) of cells [33]. Since

the metabolism is linked to the growth, it is not too surprising that existence of an

optimal size can be theoretically shown to arise from joint regulation of timing and

growth [34].

Do unicellular organisms also have an optimal size for a given growth condition?

Intuitively, a bigger bacterial cell will divide faster and thus its proliferation would

be higher than a smaller cell. However, the fact that the average bacterial cell size

grows exponentially with growth rate (per size) imposed by the nutrients suggests that

4

there may be a specific target cell size for the given environment [36], which may be

determined by several factors such as the surface to volume ratio [26], availability of

fatty acids [37], etc. Observing an optimal size in bacteria via experiment such as [32]

is perhaps hard because of a narrow range of cell sizes as compared to mammalian cells.

It would be interesting to see results of competition between mutants with artificially

large sizes, such as those obtained by expression of useless proteins in [21].

1.3 Living with size variations: gene expression homeostasis

Going beyond cell size homeostasis, single-cell approaches are also elucidating

mechanisms for gene-product concentration homeostasis. In particular, how is the con-

centration of given RNA/protein buffered to random or cell-cycle dependent fluctua-

tions in cell size. This problem is especially acute for mammalian cells, where individual

synchronized cells from the same population, exposed to identical nutrients, exhibit a

six-fold variation in size [38]. Since rates of intercellular biochemical processes typically

depend on the concentration of molecular species, cells must maintain concentration

levels despite dramatic single-cell size deviations from the population average. Adding

to size variations, unsynchronized cells also differ in the number of gene copies, and

it is unclear to what degree expression is compensated for gene dosage across organisms.

In principle, one can envision different strategies for maintaining a desired gene-

product concentration: size-dependent regulation of synthesis and/or decay rates with

perfect gene dosage compensation, or scaling of gene dosage with cell size accompa-

nied by size-independent expression per gene copy (Fig. 1.3). Prior investigations on

inferring these mechanisms have relied on bulk-assay expression measurement utilizing

cell size mutants, or altering size through small-molecules drugs [39, 40]. Instead of

artificially changing the average cell size, exploiting natural intracellular size varia-

tions within a population provides a more physiologically perturbation-free setting to

study concentration homeostasis. Building up on this theme, recent works have used

5

mRNA fluorescence in situ hybridization (RNA FISH) to count mRNAs inside indi-

vidual cell along with precise cell size measurement within a population of mammalian

cells [38,41]. For most genes, mRNA copy numbers scale linearly with size, implying a

gene-specific mRNA concentration set point (Fig. 1.4a). Intriguingly, the data further

points to gene dosage compensation - two similarly sized cells in G1 and G2 phase will

have same number of mRNAs, on average [38, 42]. While this mRNA count vs. size

scaling observed in diverse mammalian cell types seems intuitive and also previously

reported in bulk-assay experiments [40], its mechanistic underpinnings and molecular

implementations remained elusive. Measurements of promoter activity of individual

gene copies in single cells revealed for the first time that concentration homeostasis is

orchestrated through modulation of the transcriptional burst size - with increasing cell

size more RNA polymerases initiate transcription when the gene randomly switches

to an active state (Fig. 1.4b). In contrast, gene dosage compensation occurs through

the burst frequency (i.e., how often a gene become active) which is approximately

halved upon gene duplication during the S phase. It is interesting to note that sim-

ilar size-dependent transcription rates have recently been found in A. thaliana [43],

suggesting the homeostasis principles uncovered through single-cell measurements are

broadly applicable to animal and plant cells.

Studies in budding yeast show similarities and striking differences with their

mammalian counterparts. For example, as in mammalian cells, gene-dosage compen-

sation occurs during the S-phase of S. cerevisiae, wherein incorporations of acety-

lated histones into newly replicated regions leads to suppressed transcription per gene

copy [44]. However, unlike mammalian cells, nascent transcription rates for RNA poly-

merase II genes have been reported to be size-independent in budding yeast [45]. This

implies that a newborn yeast cell growing in G1 will have reduced mRNA synthesis

with increasing size, in the sense of mRNAs added per volume per unit time. Intrigu-

ingly, this decreased synthesis rate is compensated by a corresponding change in RNA

stability to maintain mRNA concentration [45]. Previously, mRNA degradation rate

modulation as a function of growth rate/temperature (which should indirectly affect the

6

cell-size) has also been observed in yeast cells [46–49]. In this case, the transcription-

degradation cross talk for concentration homeostasis is hypothesized to occur through

molecular factors that continuously shuttle between the nucleus/cytoplasm to coordi-

nate and participate in both transcription and decay processes [50]. While transcription

of RNA polymerase II genes is size-independent, transcription of ribosomal RNAs via

RNA polymerase I is size-dependent [45], and this latter mechanism maybe critical in

maintaining a fixed concentration of ribosomes.

In contrast to eukaryotic cells, the bacterial cells do not seem to compensate for

DNA dosage [51]. In this case, gene-product concentrations decrease with increase in

size for fixed ploidy, suggesting that a strong coupling between gene-dosage and cell

size can lead to near-constant gene product concentrations. In agreement with this, [52]

has shown that cyanobacteria cells are able to maintain concentration of proteins by

scaling their gene dosage with cell size. (Fig. 1.3c). A related issue is how change

in gene dosage affects concentration of its products through the cell-cycle. Intuitively,

the gene product concentration is expected to decrease until associated gene duplicates

(based on its location on the chromosome) and increase back thereafter [53]. Since genes

located at different places on the chromosome are duplicated at different times, such

imbalance in gene dosage is utilized by B. subtilis for coordination of sporulation [54].

It is also worth noting that the bacterial cell size distribution is quite narrow, hinting

that the scaling DNA dosage with cell size may suffice to maintain near constant

gene product concentrations for some genes strategically placed on the chromosome.

Recent experiments on E. coli indeed suggest that the fluctuations in protein level

are not very large (about 4% ) for several genes and a near constant concentration is

maintained during the cell-cycle [55]. These fluctuations may be further suppressed

via autoregulation which is a prevalent feedback mechanism in prokaryotes.

In the remaining chapters (Chapter 8 and 9), we discuss the conditions for

achieving cell size homeostasis, and describe how a specific example explains size reg-

ulation mechanisms in E.coli bacterium.

7

Div

isio

ns

Mother size

1

2

3

4

Ad

de

d s

ize

at d

ivis

ion

Ce

ll cycle

time

Newborn length

Siz

eBirth Division

Cell cycle

Newborn size

Added size

at G

1/S

Ad

de

d s

ize

Newborn area

E. coli

G1 d

ura

tion

Siz

e

Mammalian cell

Da

ug

hte

r

Newborn areaNewborn length

Siz

e

C. reinhardtii

Gro

wth

Newborn area

Mother size Daughter size

rate

/siz

e

Gro

wth

Newborn lengthra

te/s

ize

100 140 180

4

8

12

100 140 1800

40

80

Mother size

siz

e

a) b) c)

Birth G1/S

Fast growth

Birth G1/S

Slow growth

Birth Late G1 S/M

Figure 1.1: Diverse strategies based on regulating timing of cell-cycleevents, growth rate, and number of daughters per mother cellare exploited for maintaining cell size homeostasis. a) An E. colicell grows exponentially in size (cell length used as a proxy for size) dur-ing the cell-cycle. At the single-cell level, the cell-cycle duration sharplydecreases with increasing newborn size so as to add a fixed size from birthto division (corresponding to the Adder model; data taken from Fig. 4Aand Fig. 2F of [7]). In contrast, the growth rate (normalized by size) isuncorrelated with size (Fig. 4C of [7]). b) Unlike E. coli, mammalian cellsexhibit size-dependent growth during G1, with larger newborns growingslower as compared to small newborns. The timing of the G1 phase ex-hibits a strong negative correlation with newborn size (cell surface areaused as a proxy for size), and the correlation becomes weaker for largercells (data on Rat Basophilic Leukemia cells taken from Fig. 4C of [27]).The added size from birth to G1/S transition decreases with newborn size(corresponding to a Sizer or size-checkpoint model) for small cells, butis independent of size for larger newborns (corresponding to the Addermodel; Fig. 5A of [27]). c) The unicellular alga C. reinhardtii growsexponentially in size during the G1 period (in presence of light) and thenundergoes rapid alternating series of divisions (S phases and mitoses orS/M) to produce 2n daughters. At single-cell level, the number of divi-sion cycles n increases with mother cell size ( [28]) such that the averagedaughter cell size is held approximately constant (see Fig. 4 of [29]).

8

Fitn

ess

Size distribution

Small Intermediate Large

Cell size (FSC)

Small Intermediate Large

Cell sorting Cell culture

(Re

lativ

e c

ell

co

un

t)

Small Intermediate Large

Cell size (FSC) Cell size (FSC)

Fitness

a)

Cell size (FSC)

Fitn

ess

(Re

lativ

e c

ell

co

un

t)

72 hr

b)

Figure 1.2: An optimal cell size maximizes fitness within a population ofmammalian cells. a) Using forward scatter intensity (FSC) as a proxyfor cell size [32,35], flow cytometry is used to sort an original unsynchro-nized cell population (grey) into several subpopulations with different cellsizes. Each subpopulation is cultured for 72 hrs (approximately 3−5 cellgenerations), and fitness is quantified by measuring the relative changein cell count. Interested readers are referred to the material and methodsof [32] for further details. b) Measured fitness is plotted as a function ofthe average subpopulation FSC at the time of sorting for three differentcell types: Jurkat cells (human T lymphocyte cell line), HUVEC (humanumbilical vein endothelial cells; a primary cell line) and Kc167 (a widelyused Drosophila cell line). Original cell size distribution is shown in grey.

9

mR

NA

tra

nscri

ptio

n Haploid

Diploid

Cell size

mR

NA

ha

lf lif

e

a) b) c)

Figure 1.3: Potential mechanisms driving gene-product concentrationhomeostasis demonstrated at the RNA level. a) The rate of tran-scription (number of mRNAs synthesizes per unit time) of an individualgene increases proportionally with size in single cells, with a size-invariantdecay rate. Both haploid (light violet) and diploid (vilot) cells exhibitsimilar scaling due to gene dosage compensation. b) The transcriptionrate is independent of gene dosage and cell size, and the mRNA stability(decay rate) decreases with cell size to maintain a fixed concentration.c) The rate of transcription is size-independent and increases by approx-imately 2-fold upon gene replication. A strong coupling between cell sizeand gene dosage leads to concentration homeostasis. Unlike the first twostrategies, here homeostasis is not perfect as mRNA concentrations willdecrease with increasing size for fixed ploidy.

10

mR

NA

de

ca

y c

on

sta

nt

Gene off Gene on RNA polymerase mRNA

0

1

2

0

0.5

1.0

0 2 4 6 80

0.1

0.2

0.3

0

2

mR

NA

co

un

t

Cell size (picoliters)

IER2

UBCa) b)

Time

Figure 1.4: Mammalian cells maintain mRNA concentrations independentof size through modulation of transcriptional burst size and fre-quency with cell size. a) The mRNA copy numbers measured via RNAFISH for two genes (UBC and IER2) scale linearly with size across singlecells from the same population, while the mRNA half-lives are indepen-dent of size (Fig. 3A and 3B of [38]). Both chosen mRNAs are relativelyunstable, and hence mRNA count is a proxy for de novo transcriptionrate. b) Schematic of a promoter switching between transcriptionally in-active and active states. The linear scaling of transcription rate with sizeresults from a higher burst size (the number of mRNAs synthesized fromthe active state) in larger cells. Upon gene replication, the fraction oftime the promoter in ON is approximately halved leading to gene dosagecompensation [38]. As a consequence, two similarly-sized cells in G1 andG2 will have the same mRNA count.

11

Chapter 2

CONDITIONS FOR CELL SIZE HOMEOSTASIS: A STOCHASTICHYBRID SYSTEMS APPROACH

Stochastic hybrid systems (SHS) constitute an important mathematical mod-

eling framework that combines continuous dynamics with discrete stochastic events.

Here we use SHS to model a universal feature of all living cells: growth in cell size

(volume) over time and division into two viable progenies (daughters). A key ques-

tion is how cells regulate their growth and timing of division to ensure that they do

not get abnormally large (or small). This problem has ben referred to literature as

size homeostasis and is a vigorous area of current experimental research in diverse

organisms [3, 7, 8, 11,18,59–69]. We investigate if phenomenological models of cell size

dynamics based on SHS can provide insights into the control mechanisms needed for

size homeostasis.

The proposed model consists of two non-negative state variables: v(t), the size

of an individual cell at time t, and a timer τ that measures the time elapsed from

when the cell was born (i.e., last cell division event). This timer can be biologically

interpreted as an internal clock that regulates cell-cycle processes. Time evolution of

these variables is governed by the following ordinary differential equations

v = α(v, τ )v, τ = 1, (2.1)

where the growth rate α(v, τ ) ≥ 0 can depend on both state variables and is such that

(2.1) has a unique and well-defined solution ∀t ≥ 0 (i.e., cell size does not blow up in

finite time). A constant α implies exponential growth over time.

As the cell grows in size, the probability of cell division occurring in the next

infinitesimal time interval (t, t + dt] is given by f(v, τ )dt, where f(v, τ ) can be inter-

preted as the division rate. Whenever a division event is triggered, the timer is reset to

12

zero and the size is reduced to βv, where random variable β ∈ (0, 1) is drawn from a

beta distribution. Assuming symmetric division, β is on average half, and its coefficient

of variation (CVβ) quantifies the error in partitioning of volume between daughters.

To be biologically meaningful, α(v, τ ) is a non-increasing function, while f(v, τ ) is a

non-decreasing function of its arguments. The SHS model is illustrated in Fig. 2.1 and

incorporates two key noise sources: randomness in partitioning and timing of division.

Next, we explore conditions for size homeostasis, in the sense that, the mean cell size

does not converge to zero, and all statistical moments of v remain bounded.

Figure 2.1: SHS model for capturing time evolution of cell size. The size of anindividual cell v(t) grows exponentially with growth rate α(v, τ ), whereτ represents a timer that measures the time since the last division event.The arrow represents cell division events that occur with rate f(v, τ ),which resets τ to zero and divide the size by approximately half. Asample trajectory of v(t) is shown with cycles of growth and division.

2.1 Timer-dependent growth and division

We begin by considering a scenario, where both the growth and division rates

are functions of τ , but do not depend on v. The SHS can be compactly written as

v = α(τ )v, τ = 1, (2.2)

with reset maps

v 7→ βv, τ 7→ 0 (2.3)

that are activated at the time of division. The timer-controlled division rate f(τ )

can be interpreted as a “hazard function” [70]. Let T1, T2, . . . denote independent

13

and identically distributed (i.i.d.) random variables that represent the time interval

between two successive division events. Then, based on the above formulation, the

probability density function (pdf) for Ti is given by

Ti ∼ f(x)e−∫ xy=0 f(y)dy, ∀x ≥ 0 (2.4)

[70]. Note that a constant division rate in (2.4) would lead to an exponentially dis-

tributed Ti. For this class of models, the steady-state statistics of v is given by the

following theorem.

Theorem 1: Consider the SHS (2.2)-(2.3) with timer-dependent growth and division

rates. Then

limt→∞〈v(t)〉 =

0⟨e∫ Tiy=0 α(y)dy

⟩< 2

∞⟨e∫ Tiy=0 α(y)dy

⟩> 2,

(2.5)

where the symbol 〈 〉 is used to denote the expected value of a random variable. More-

over,

0 < limt→∞〈v(t)〉 <∞, lim

t→∞〈v2(t)〉 =∞ (2.6)

when⟨e∫ Tiy=0 α(y)dy

⟩= 2. �

Proof of Theorem 1: Let vi−1 denote the cell size just at the start of the ith cell

cycle. Using (2.2), the size at the time of division in the ith cell cycle is given by

vi−1e∫ Tiy=0 α(y)dy. (2.7)

Thus, the size of the newborn cell in the next cycle is

vi = vi−1xi, xi := βie∫ Tiy=0 α(y)dy, (2.8)

where βi ∈ (0, 1) are i.i.d random variables following a beta distribution and xi are

i.i.d. random variables that are a function of βi and Ti. From (2.8), the mean cell size

at the start of ith cell cycle is given by

〈vi〉 = v0〈xi〉i (2.9)

14

and will grow unboundedly over time if 〈xi〉 > 1, or go to zero if 〈xi〉 < 1. Using the

fact that 〈βi〉 = 0.5 (symmetric division of a mother cell into daughter cells), βi and Ti

are independent, (2.5) is a straightforward consequence of (2.9). It also follows from

(2.8) that

〈v2i 〉 = v2

0〈x2i 〉i = v2

0〈xi〉2i(1 + CV 2xi

)i (2.10)

where CV 2xi

represents the coefficient of variation squared of xi. When 〈xi〉 = 1 then

〈vi〉 = v0 and

〈v2i 〉 = v2

0(1 + CV 2xi

)i. (2.11)

Note that when the system is completely deterministic, i.e., pdfs for Ti and βi are given

by delta functions, CV 2xi

= 0. However, the slightest noise in these variables will lead

to CV 2xi> 0, in which case (2.11) implies (2.6). �

In summary, unless functions α(τ ) and f(τ ) are chosen such that⟨e∫ Tiy=0 α(y)dy

⟩=

2, the mean cell size would either grow unboundedly or go extinct. Moreover, even

if the mean cell size converges to a non-zero value, the statistical fluctuations in size

would grow unboundedly. Hence, size-based regulation of growth/division rates is a

necessary condition for size homeostasis .

2.2 Size-dependent growth rate

Recent work measuring sizes of single mammalian cells over time has reported

lowering of growth rates as cells become bigger [71–73]. To explore the effects of such

regulation, we consider a growth rate α(v, τ ) that now depends on size. As in the

previous section, timer-controlled division events occur with rate f(τ ) resulting in

inter-division times Ti given by (2.4). The following result shows that size homeostasis

is possible if growth rate is appropriately bounded from below and above.

Theorem 2: Let the growth rate be bounded by

α(v, τ )v ≤ k(τ )vp, p ∈ [0, 1), ∀v ≥ 0 (2.12)

15

for some non-increasing function k(τ ). Moreover, the growth rate of a small cell is

large enough such that⟨e∫ Tiy=0 α0(y)dy

⟩> 2, α0(τ ) := lim

v→0α(v, τ ). (2.13)

Then

0 < limt→∞〈vl(t)〉 <

(l〈k(τ )〉〈Ti〉〈1− βl〉

) 11−p

(2.14)

where l ∈ {1, 2, . . . }, 〈Ti〉 is the mean cell-cycle duration, and β ∈ (0, 1) is a random

variable quantifying the error in partitioning of volume between daughters. �

Proof of Theorem 2: Consider a newborn cell with a sufficiently small size born at

time t = 0. Then, the mean cell size will grow in successive generation iff the second

inequality in (2.5) is true for α0(τ ), which results in (2.13). Based on the Dynkin’s

formula for the SHS (2.1) and (2.3), the time evolution of moments is given by

d〈vl〉dt

=⟨f(τ )vl

⟩ (〈βl〉 − 1

)+ l⟨α(v, τ )vl

⟩, (2.15)

for l ∈ {1, 2, . . . } [74]. Using (2.12),

d〈vl〉dt≤⟨f(τ )vl

⟩ (〈βl〉 − 1

)+ l⟨k(τ )vl−1+p

⟩. (2.16)

Note that ⟨f(τ )vl

⟩=⟨f(τ )〈vl|τ 〉

⟩(2.17)

where 〈vl|τ 〉 is the expected value of vl conditioned on τ . Based on the time evolution

of cell size in (2.1), 〈vl|τ 〉 is an increasing function of τ (cells further along in the

cell cycle, have on average, larger sizes). Since 〈vl|τ 〉 and f(τ ) are monotone non-

decreasing function of τ ⟨f(τ )vl

⟩≥ 〈f(τ )〉〈vl〉. (2.18)

Similarly, since k(τ ) is a non-increasing function,

⟨k(τ )vl−1+p

⟩≤ 〈k(τ )〉〈vl−1+p〉. (2.19)

16

Finally, using the fact that l − 1 + p ≤ l as p ∈ [0, 1)

〈vl−1+p〉 =

⟨(vl) l−1+p

l

⟩≤⟨vl⟩ l−1+p

l (2.20)

Using (2.18)-(2.20), (2.16) reduces to the following inequality

d〈vl〉dt≤〈f(τ )〉〈vl〉

(〈βl〉 − 1

)+ l〈k(τ )〉

⟨vl⟩ l−1+p

l . (2.21)

Since at steady state

〈f(τ )〉 =1

〈Ti〉, (2.22)

[75], (2.21) implies (2.14). �

An extreme example of size-dependent growth is

α(v, τ ) =k

v, k > 0 (2.23)

which corresponds to cells growing linearly in size, as experimentally reported for some

organisms [76]. For this case, the result below provides exact closed-form expressions

for the first and second-order statistical moments of v.

Theorem 3: Consider the growth rate (2.23) that results in the following SHS contin-

uous dynamics

v = k, τ = 1. (2.24)

Then, the steady-state mean and coefficient of variation squared of cell size is given by

limt→∞〈v(t)〉 =

k〈Ti〉(3 + CV 2

Ti

)2

, (2.25)

CV 2v =

1

27+

4(

9〈T 3

i 〉〈Ti〉3 − 9− 6CV 2

Ti− 7CV 4

Ti

)27(3 + CV 2

Ti

)2

+16CV 2

β

3(3− CV 2β )(3 + CV 2

Ti), (2.26)

where CV 2Ti

and CV 2β denote randomness in the inter-division times (Ti) and parti-

tioning errors (β), respectively, as quantified by their coefficient of variation squared.

�

17

The proof of Theorem 3 can be found in the Appendix. Interestingly, the mean

cell size in (2.25) not only depends on the mean inter-division times 〈Ti〉, but also on

its second-order moment CV 2Ti

. Thus, making the cell division times more random (i.e.,

increasing CV 2Ti

) will also lead to larger cells on average. Similar effects of CV 2Ti

on

mean gene expression levels have recently been reported in literature [77, 78]. More-

over, (2.26) shows that the magnitude of fluctuations in cell size (CV 2v ) depend on

Ti through its moments up to order three. Note that if CV 2β = 0 (no partitioning

errors) and Ti = 〈Ti〉 with probability one (deterministic inter-division times), then

CV 2v = 1/27. This non-zero value for CV 2

v in the limit of vanishing noise sources repre-

sent variability in size from cells being in different stages of the deterministic cell cycle.

Theorem 3 decomposes CV 2v into terms representing contributions from different noise

sources. The terms from left to right in (2.26) represent contributions to CV 2v from i)

Deterministic cell-cycle and ii) Random timing of division events and iii) Partitioning

errors at the time of division. Assuming lognormally distributed Ti,

〈T 3i 〉/〈Ti〉3 =

(1 + CV 2

Ti

)3. (2.27)

Substituting (2.27) in (2.26) and plotting CV 2v as a function of CV 2

β and CV 2Ti

, re-

veals that stochastic variations in cell size are more sensitive to partitioning errors as

compared to noise in the inter-division times.

In summary, our result show that appropriate regulation of growth rate by

size (as seen in mammalian cells) can be an effective mechanism for achieving size

homeostasis. We next consider a different class of models where size-based regulation

is at the level division rather than growth.

2.3 Size-dependent division rate

In contrast to growth rate control, many organisms rely on size-dependent reg-

ulation of division rate for size homeostasis [2, 3, 79–81]. To analyze this strategy, we

consider the SHS continuous dynamics (2.2) with a timer-dependent growth rate α(τ ),

and a division rate f(v, τ ) that now depends on size. The theorem below provides

18

Noise in partitioning and cell cycle time

Me

an c

ell

siz

e

Sto

chastic v

ari

abili

ty

in c

ell

siz

e

Cell cycle time

Partitioning

Figure 2.2: Stochastic variation in cell size (blue) and mean cell size (green) as a func-tion of CV 2

Ti(noise in inter-division time) and CV 2

β (error in partitioningof volume among daughters) for linear cell growth and a timer-baseddivision mechanism. The mean cell size is dependent on CV 2

Tibut in-

dependent of CV 2β . Fluctuations in cell size increase more rapidly with

CV 2β than with CV 2

Ti.

19

sufficient conditions on f(v, τ ) for size homeostasis.

Theorem 4: Let there exist a non-decreasing function g(τ ) and p > 0 such that

f(v, τ ) ≥ g(τ )vp. (2.28)

Moreover, the division rate for a sufficiently small cell size f0(τ ) := limv→0 f(v, τ )

satisfies ⟨e∫ Tiy=0 α0(y)dy

⟩> 2, Ti ∼ f0(x)e−

∫ xy=0 f(0y)dy. (2.29)

Then, for the SHS given by (2.2) and (2.3)

0 < limt→∞〈vl(t)〉 <

(l〈α(τ )〉

〈g(τ )〉(1− 〈βl〉)

) lp

, (2.30)

for l ∈ {1, 2, . . . }. �

Proof of Theorem 4: Consider a newborn cell with a sufficiently small size at time

t = 0. Then, based on Theorem 1, the mean size will grow over successive generations

(and not go extinct) iff (2.29) holds. Based on the Dynkin’s formula for (2.2)-(2.3),

the time evolution of moments is given by

d〈vl〉dt

=⟨lα(τ )vl

⟩−⟨f(v, τ )vl

⟩ ⟨1− βl

⟩(2.31)

Using (2.28), the fact that α(τ ) is a non-increasing function, while g(τ ) is a non-

decreasing function,

d〈vl〉dt≤l 〈α(τ )〉

⟨vl⟩− 〈g(τ )〉

⟨vl+p

⟩ ⟨1− βl

⟩(2.32)

Finally, using⟨vl+p

⟩≥⟨vl⟩ l+p

l in (2.32) result in (2.30) at steady state. �

Next, we show that different known strategies for size-dependent regulating

of inter-division times are consistent with Theorem 4. A common example of size-

dependent division is the “sizer strategy”, where a cell senses its size, and divides when

20

a critical size threshold is reached [24,82–84]. Such as strategy can be implemented by

f(v, τ ) =(vv

)p(2.33)

where v and p are positive constant. A large enough p corresponds to division events

occurring when size reaches v. In contrast to the sizer strategy, many bacterial species

use an “adder strategy”, where a cell divides after adding a fixed size from birth

[6, 9, 10, 19]. In the case of exponential growth (constant growth rate α), the adder

strategy can be implemented by

f(v, τ ) =

(v (1− e−ατ )

v

)p. (2.34)

A large enough p would correspond to cells adding a fixed size v between cell birth

and division [22]. Both these division rates are consistent with the form of f required

for size homeostasis in Theorem 4. We investigate the first two moments of v in more

detail for the sizer strategy.

Using (2.31) for a constant growth rate α and division rate (2.33) results in the

following moment dynamics

d⟨vl⟩

dt= lα

⟨vl⟩− v−p

⟨vl+p

⟩ ⟨1− βl

⟩. (2.35)

Let µ =[〈v〉 , 〈v2〉 · · ·

⟨vL⟩]T

be a vector of moments up to order L, where L is the

order of truncation. Using (2.35), the time evolution of µ can be compactly written as

dµ

dt= a+ Aµ+ Cµ, µ =

[⟨vL+1

⟩· · ·⟨vL+p

⟩]T(2.36)

for some vector a, matrices A and C, and µ is the vector of higher order moments.

Note that nonlinearities in the division rate lead to the well known problem of moment

closure, where time evolution of µ depends on higher-order moments µ. Moment closure

techniques that express µ ≈ θ (µ) are typically used to solve equations of the form

(2.36). Here, we use closure schemes based on the derivative-matching technique [85–

87], that yield analytical expressions for the steady-state moments. For example, L = 2

21

in (2.36) (second order of truncation) results in the following steady-state mean and

coefficient of variation squared of cell size

〈v〉 ≈ 21pα

1p v

(3− CV 2

β

4

) p+12p

, CV 2v ≈

(4

3− CV 2β

) 1p

− 1, (2.37)

respectively. Intriguingly, (2.37) shows that the mean cell size decreases with increasing

magnitude of partitioning error CV 2β . While the results from (2.37) are qualitatively

consistent with moments obtained via Monte Carlo simulations, a much higher order

of truncation is needed in (2.36) to get an exact quantitative match (Fig. 3).

2nd order

20th order

Approximation:

Simulations

Figure 2.3: Stochastic variation in cell size (blue) and mean cell size (green) as afunction of CV 2

β (error in partitioning of volume among daughters) forexponential cell growth and sizer-based division mechanism. The meancell size decreases with increasing CV 2

β , while noise in cell size increases

with it. Results are shown for a 2nd (dashed) and a 20th (solid) ordermoment closure truncation, and compared with moments obtained byrunning a large number of Monte Carlo simulations. Errors bars show95% confidence estimates.

Here we have used a phenomenological SHS framework to model time evolution

of cell size (Fig. 8.1). The model is defined by three features: a growth rate α(v, τ ), a

22

division rate f(v, τ ), and a random variable β ∈ (0, 1) that determines the reduction

in size when division occurs. A key assumption was that α and f are monotone

functions: with increasing size and cell-cycle progression, the growth rate decreases,

and propensity to divide increases. Our main contribution was to identify sufficient

conditions on α and f that prevent size extinction and also lead to bounded moments

(Theorems 2 and 4). In essence, these conditions require the growth (division) rate to

decrease (increase) with cell size in a polynomial fashion.

We also analyzed two strategies for size homeostasis: i) Linear growth in size

with timer-controlled divisions and ii) Exponential growth in size with size-controlled

divisions. Analysis reveals that in the former strategy, the mean cell size is independent

of volume partitioning errors at the time of mitosis. In contrast, the mean cell size

decreases with increasing partitioning errors for size-controlled divisions. Moreover,

stochastic variations in cell size are found to be highly sensitive to partitioning errors

for both strategies (Fig. 2 and 3). This suggests that cells may use mechanisms to

minimize volume mismatch among daughter cells. In summary, theoretical tools for

SHS can provide fundamental understanding of regulation needed for size homeostasis.

Future work will focus on coupling cell size to gene expression, and understanding how

concentration of a given protein is maintained in growing cells [38,40,88,89].

23

Chapter 3

A MECHANISTIC STOCHASTIC FRAMEWORK FOR REGULATINGBACTERIAL CELL DIVISION

Recurring cycles of growth and division of a cell is a ubiquitous theme across all

organisms. How an isogenic population of exponentially growing cells maintains a nar-

row distribution of cell size, a property known as size homeostasis, has been extensively

studied, e.g., see [1, 3, 83, 90] and references therein. From a phenomenological stand-

point, recent experiments reveal that diverse microorganisms achieve size homeostasis

via an adder principle [7–10]. As per this strategy, cells add a constant size from birth

to division regardless of their size at birth [6,91]. Interestingly, the size accumulated by

a single cell between birth and division exhibits considerable cell-to-cell differences, and

these differences follow unique statistical properties. For example, in a given growth

condition, the added size is drawn from a fixed probability distribution independent

of the newborn cell size. Moreover, the distribution of the added size normalized by

its mean is invariant across growth conditions [8]. Here, we explore biophysical mod-

els that lead to the adder principle of cell size control and provide insights into its

statistical properties.

To realize the adder principle mechanistically, a cell needs to somehow track the

size it has accumulated since the previous division and trigger the next division upon

addition of the desired size. One biophysical model proposed to achieve this assumes

a protein which begins to get expressed right after cell birth at a rate proportional to

instantaneous volume (size). The cell grows exponentially over time and division is

triggered when protein copy numbers reach a critical threshold after which the protein

is assumed to degrade (Fig. 3.1a) [6, 9, 21]. Such copy number dependent triggering

of cell division could potentially be implemented via the localization of protein into

24

compartments whose volume does not change appreciably with the cell volume [62].

Moreover, the synthesis and the degradation of the protein in this model are used in

broad sense; they could as well be activation of timekeeper proteins in size dependent

manner, and deactivation after triggering of division. While this deterministic model

results in a constant size added from cell birth to division [6, 21], it remains to be

seen how noise mechanisms can be incorporated in this model to explain statistical

fluctuations in cell size. A plausible source of noise could be the inherent stochastic

nature of protein expression that has been universally observed in prokaryotes and

eukaryotes [92–96]. Such stochasticity in protein synthesis is amplified at the level of

individual cells, where gene products are often present at low molecular counts.

Considering noisy expression of the timekeeper protein, one can formulate cell-

division time as a first-passage time problem: an event (division) occurs when a stochas-

tic process (protein copy numbers) hits a threshold for the first time (Fig. 3.1b). Ex-

ploiting this first-passage time framework, we derive an exact analytical formula for

the cell-division time distribution for a given newborn cell size. Consistent with data,

these results predict that the mean cell-division time decreases with increasing cell size

at birth, and the randomness (quantified by coefficient of variation squared) in the

cell-division time increases with newborn cell size. Intriguingly, analysis of the model

further shows that the distribution of the volume added from cell birth to division is

always independent of the newborn cell size. Finally, we find that the distributions

of added volume and cell division time have scale invariant forms: distributions in

different growth conditions collapse upon each other after rescaling them with their

respective means. We discuss potential candidates for the timekeeper protein and

deliberate upon model modifications that result in deviations from the adder principle.

Consider a newborn cell with volume Vb at time t = 0. Its volume at a time t

after birth is given by V (t) = Vb exp(αt), where α > 0 represents the growth rate. After

cell birth, the timekeeper protein begins to get transcribed at a rate r(t) = kmV (t),

where km is the transcription rate in the concentration sense. Note that this scaling

of protein synthesis with instantaneous cell volume is essential for preserving gene

25

product concentrations in growing cells. In the stochastic formulation, the probability

of a transcription event occurring in an infinitesimal time interval (t, t + dt] is given

by r(t)dt. Assuming short-lived mRNAs, each transcript degrades instantaneously

after producing a burst of protein molecules [97–102]. Stochastic expression of the

timekeeper protein is compactly represented by the following biochemical reaction:

∅ r(t)−−→ Bi × Protein, (3.1)

where r(t) = kmV (t) can be interpreted as the burst arrival rate and Bi, i ∈ {1, 2, · · · },

are identical and independent random variables denoting the size of protein bursts

with mean b := 〈Bi〉. The burst size represents the number of protein molecules

synthesized in a single mRNA lifetime and typically follows a geometric distribution

[98, 100, 102–105]. However, to allow a wide range of protein accumulation processes

to be covered by equation (3.1), we assume that Bi follows an arbitrary non-negative

integer-valued distribution. One example of such a mechanism could be to consider a

protein A whose concentration is constant throughout the cell cycle. This protein is

stochastically converted to an active form A∗ at a rate proportional to the number of

molecules of A. In essence, this can be thought of as production of A∗ in bursts which

takes place at a rate proportional to the cell volume.

Let x(t) denote the number of timekeeper molecules in the cell at time t af-

ter birth. Assuming a stable protein with no active proteolysis, we have x(t) =∑ni=1Bi, x(0) = 0, where n is the number of bursts (transcription events) in [0, t].

Cell division occurs when x(t) reaches a threshold X and the protein is degraded (or

deactivated) thereafter. Given this timing mechanism, cell-division time can be math-

ematically represented as the first-passage time (FPT )

FPT := inf {t : x(t) ≥ X |x(0) = 0} . (3.2)

This first-passage time framework assumes that cell division occurs upon precise

attainment of X protein molecules. In principle, one could generalize equation (3.2)

by defining a monotonically increasing function h(x) that defines a probabilistic rate

26

Figure 3.1: Proposed molecular mechanism to realize adder principle of cell size con-trol. (a) An exponentially growing rod-shaped cell starts synthesizinga timekeeper protein after its birth. The production rate of the proteinscales with the cell size (volume). When the protein’s copy number at-tains a certain level, the cell divides and the protein is degraded. (b)Stochastic evolution of the protein copy numbers is shown for cells ofthree different sizes at birth. The threshold for triggering cell division isassumed to be 50 molecules. The distribution of the first-passage time(generated via 1, 000 Monte Carlo realizations) for each newborn cellvolume is shown above the three corresponding trajectories. The first-passage time distribution depends upon the newborn cell size: on average,the protein in a smaller cell takes more time to reach the threshold ascompared to the protein in a larger cell.

of cell division at time t given x(t) molecules. Interestingly, analysis reveals that the

average size added from birth to division is invariant of the newborn cell size Vb iff

h(x) = 0 for x < X, h(x) =∞ for x > X (3.3)

(see Supplementary Information (SI), section S1). Thus, a sharp threshold, where cell

division cannot be triggered before attainment of a precise number of molecules seems

to be a necessary ingredient of the adder principle.

27

3.1 Distribution of the cell-division time given newborn cell size

Here we derive the distribution of the cell-division time (FPT ) for a given

newborn cell size Vb and investigate how its statistical moments depend on Vb. We

begin by finding the distribution of the minimum number of burst events N required

for x(t) to reach the threshold X. In particular,

N := inf

{n :

n∑i=1

Bi ≥ X

}=⇒ Prob (N ≤ n) = Prob

(n∑i=1

Bi ≥ X

). (3.4)

Given a specific form for the distribution of Bi, the corresponding distribution for N

can be obtained using equation (3.4). For example, if Bi is geometrically distributed,

then the probability mass function of N is given by

fN(n) := Prob (N = n) =

(n+X − 2

n− 1

)(1

b+ 1

)n−1(b

b+ 1

)X, n ∈ {1, 2, . . .},

(3.5)

where b represents the mean burst size [106,107].

Having determined the number of bursts needed for cell division, we next focus

on the timing of burst events. Let Tn represent the time at which nth burst event

takes place. If the burst arrival rate in equation (3.1) were constant, then the time

intervals between bursts would be exponentially distributed, resulting in an Erlang

distribution for Tn. However, in our case this rate is time varying (due to dependence

on cell volume), the arrival of bursts is an inhomogeneous Poisson process. Employing

the distribution for the timing of the nth event, and using the fact that FPT is same

as the time at which the N th burst event occurs, the probability density function of

FPT is obtained as

fFPT (t) =∞∑n=1

fTn (t) fN(n) =∞∑n=1

(R(t))n−1

(n− 1)!r(t) exp(−R(t))fN(n), (3.6)

R(t) :=

∫ t

0

r(s)ds =kmVbα

(eαt − 1

), (3.7)

(see SI, section S2). One can note that fFPT (t) is dependent on the newborn cell size

Vb through the function R(t).

28

Figure 3.2: Both model prediction and data show increase in the noise in timing asnewborn cell size increases. (a) Model prediction for noise (coefficientof variation squared, CV 2) in division time as computed numericallyusing equation (3.7) . The model parameters used are: transcriptionrate km = 0.13 min−1, threshold X = 65 molecules, growth rate α =0.03 min−1, and mean burst size b = 5 molecules. The distribution ofprotein burst size Bi is assumed to be geometric. For details on how theseparameter values were estimated, see SI, section S6. (b) Experimentaldata from [90] for Escherichia coli MG1655 also shows increase in celldivision time noise as newborn cell size increases. Single-cell data wascategorized in one of the four bins (1−2.8 µm, 2.8−4.5 µm, 4.5−6.3 µm,and 6.3−8 µm) depending upon newborn cell sizes. CV 2 of division timewith 95% confidence interval (using bootstrapping) for each bin is shown(more details in SI, section S6).

This FPT distribution qualitatively emulates the experimental observations

that the mean cell division time decreases with increasing cell size at birth (see SI,

section S6). Intuitively, a larger newborn cell expresses the protein at a higher rate

as compared to a smaller cell. Hence, the time taken by the protein to reach the

prescribed molecular threshold is shorter in larger cells. Analysis of equation (3.7) also

predicts that the noise (quantified using the coefficient of variation squared, CV 2) in

cell-division timing increases with increasing Vb, and we confirmed this behavior from

published data (Fig. 3.2). The noise behavior can be understood from the fact that a

small newborn cell requires more time for cell division. This allows for efficient time

averaging of the underlying bursty process resulting in lower stochasticity in FPT .

29

3.2 Distribution of the volume added between divisions

(a) (b)

(c) (d)Experimental data

Me

an

siz

e a

dd

ed

Me

an

siz

e a

dd

ed

Time (minutes)

Time (minutes)

Siz

e

Simulations

Figure 3.3: The proposed mechanism results in added cell size distribution being inde-pendent of the newborn cell size. (a) The cell volume grows exponentially(shown for three different newborn cell sizes) until the timekeeper pro-tein reaches a critical threshold. (b) The size added to the newborn cellsize also grows exponentially until division takes place. For three differ-ent newborn cell sizes, the distribution of the the added volume comesout to be same. (c) The added size generated via simulations is plottedagainst the newborn cell size in range 2 − 3.5 µm for 10, 000 cells. Thecells are further binned in 13 uniformly spaced bins (number of cells perbin > 100). The dashed line shows the mean of the added volume, whichis independent of the newborn cell size. (d) Data from [8] showing theadded size versus newborn cell size for Escherichia coli NCM3722 grownin Glucose as carbon source. Cells were categorized into bins accordingto their newborn cell size (number of cells per bin > 100). For each bin,the circle shows mean of the added size whereas the error bar representsthe standard deviation of the added size. It can be seen that the meanadded cell size (shown by dashed line) is independent of the newborn cellsize (also see Fig. 2D in [8]).

Having derived the distribution for the cell-division time (FPT ), we determine

30

the volume added by a single cell from birth to division (denoted by ∆V ). Since

volume grows exponentially, ∆V is related to FPT as ∆V = Vb(eαFPT − 1

). Using

the distribution of FPT from equation (3.7) yields the following probability density

function for ∆V

f∆V (v) =∞∑n=1

(kmvα

)n−1

(n− 1)!

kmα

exp

(−kmv

α

)fN(n) (3.8)

(see SI, section S3). One striking observation is that f∆V (v) is independent of the

initial volume Vb (as illustrated in Fig. 3.3). This is in agreement with experimental

observations that the histograms of the added volume for different newborn cell sizes

are statistically identical [8]. Next, we investigate how statistical moments of ∆V

depend on model parameters, in particular, the growth rate α.

Mean volume added between divisions

Using equation (3.8), the average volume added is obtained as

〈∆V 〉 =

∫ v=∞

v=0

vf∆V (v)dv =∞∑n=1

α n

kmfN(n) =

α

km〈N〉 . (3.9)

Here 〈N〉 represents the mean number of protein burst events from cell birth to division,

which depends on the threshold X and the form of the burst size distribution. For

example, if the protein bursts Bi are geometrically distributed with mean b, then using

equation (3.5)

〈∆V 〉 =α

km

(X

b+ 1

). (3.10)

These formulas reveal a linear dependence of ∆V on α, in agreement with data from

Pseudomonas aeruginosa [9]. It turns out that the dependency of ∆V on α can vary

among bacterial species. For instance, Caulobacter crescentus exhibits an added vol-

ume independent of α, whereas this relationship is thought to be exponential in case

of Escherichia coli [7, 8]. Studies connecting cellular growth rates to gene expression

parameters have shown that α primarily affects the transcription rate, with mRNA

translation and stability being largely invariant across growth conditions [108, 109].

31

Thus, if the transcription rate km is a linear function of α, then ∆V becomes inde-

pendent of α. Next, we discuss a slightly different model formulation that results in

exponential dependency of ∆V on α.

So far we have considered that the timekeeper protein observes time from cell

birth to division. In principle, the timekeeping could be for some other important

event in the cell cycle. Consider a scenario where the initiation of DNA replication

takes place when sufficient timekeeper protein has accumulated per origin of replica-

tion [6, 13, 110, 111]. The corresponding division event is assumed to occur with a

constant delay of T after an initiation. The delay T here is the C + D period, where

C represents the time to replicate the DNA and D denotes the time between DNA

replication and division [112, 113]. As growing bacterial cells are known to regulate

the number of DNA replication forks as a function of growth rate, we assume that the

threshold for the timekeeper proteins changes accordingly. More specifically, if there

are θ origins of replication, the number of timekeeper protein molecules required to

be accumulated for the next initiation event are θX. The above assumption is consis-

tent with the understanding that all origins of replication fire almost synchronously.

Further, the timekeeper molecules are assumed to get degraded (deactivated) after

initiation and a new set of timekeeper molecules are produced for the next initiation.

Upon a division event between two successive initiations, the partitioning errors in the

timekeeper protein are assumed to be negligible.

In this alternative formulation, the average volume added between two con-

secutive initiation events for each origin of replication is approximately same as ∆V

obtained in equation (3.10) (see SI, section S3). Moreover, the average volume added

between successive division events is now given by [13]

〈∆V ∗〉 ≈ 〈∆V 〉 eαT . (3.11)

Recall from equation (3.10) that 〈∆V 〉 depends linearly on α. Thus, the expression

in equation (3.11) suggests two different regimes of how 〈∆V ∗〉 depends upon α. For

small values of α, α exp(αT ) ≈ α, i.e., the mean added volume increases linearly with

32

the growth rate. In the regime where α is large, the exponential term dominates. This

implies that if α is small, it may not be possible to distinguish whether the underlying

mechanism accounts for volume added between two division events or two initiation

events as the data will show a linear dependence of the average added volume with

changes in α [9]. Notice that a pure exponential relationship between 〈∆V ∗〉 and α

can also be obtained if km is a linearly increasing function of α. For this particular

case, the volume accounted by each origin of replication 〈∆V 〉 becomes invariant of

the growth rate, consistent with previous works [13, 114]. In summary, depending on

the underlying assumptions, the model captures a variety of relationships between the

average volume added from cell birth to division and α.

It is noteworthy that in the above setup, dependency of the time T = C+D on

growth rate or cell size has been neglected even though there is evidence that D usually

depends upon both growth rate and cell size [17]. We have done so for simplicity as

incorporating this would not change the fact that an exponential dependency can be

generated between ∆V and α by having the protein account for two other events in

the cell cycle. We next investigate higher order moments of ∆V in the original model

formulation, where the timekeeper protein accounts for timing between division events.

3.3 Higher order moments of added volume

We can use the distribution of ∆V computed in equation (3.8) to get insights

into its higher-order statistics such as coefficient of variation squared (CV 2∆V ) and

skewness (skew∆V ). For example, when the protein production occurs in geometric

bursts

CV 2∆V =

b2 + 2bX +X

(b+X)2, skew∆V =

2 (b3 + 3b2X + 3bX +X)

(b2 + 2bX +X)3/2(3.12)

(see SI, section S3). Note that ∆V is always positively skewed, consistent with previous

understanding [91]. Moreover, both CV 2 and skewness are independent of the growth

rate α. It turns out an even more general property is true: an appropriately scaled jth

order moment of ∆V , i.e., 〈∆V j〉 / 〈∆V 〉j is independent of α, in spite of the underlying

33

distribution of the burst size. This arises from the fact that the distribution of ∆V

can be written in the following form

f∆V (v) =1

〈∆V 〉g

(v

〈∆V 〉

)(3.13)

for some function g (see SI, section S3). This form implies that f∆V (v) is scale invariant:

the shape of the distribution across different growth rates is essentially the same, and

a single parameter 〈∆V 〉 is sufficient to characterize the distribution of ∆V [115]. This

property was seen in experiments [8, 21, 116], where the histograms for ∆V/ 〈∆V 〉 in

different growth conditions collapse upon each other (Fig. 3.4).

Figure 3.4: Collapse of added cell size in different growth conditions upon rescalingby respective mean values. (a) Using data from [8] for Escherichia coliNCM3722, the added size is plotted versus the newborn cell size fordifferent growth conditions. The mean added size (shown by circles) foreach growth condition is different for a given newborn cell size. Cellswere categorized into bins according to their newborn cell sizes (numberof cells per bin > 100). The error bars represent the standard deviationof the added volume of cells in each bin. (b) The added size data fordifferent growth conditions collapse upon rescaling them by their meansin the respective growth conditions (also see Fig. 2D in [8]).

Interestingly, the above invariance property is not limited to the distribution of

the added volume ∆V . As the distributions of the cell size at birth, and cell size at

34

division are generated by weighted sums of random variables drawn from the distri-

bution of ∆V , they naturally inherit the scale-invariance property [8] (see SI, section

S4). Furthermore, the distribution of the cell-division time also has the scale invariance

property (see SI, section S5), which is in agreement with previous works [117,118].It is now well understood that several prokaryotes, such as, Escherichia coli,

Caulobacter crescentus, Bacillus subtilis and Pseudomonas aeruginosa employ an addermechanism for size homeostasis [7–10]. In this work, we studied a simple molecularmechanism for realizing the adder principle that consists of a timekeeper protein ex-pressed at a rate proportional to cell volume up to a critical threshold. Our workshows that stochastic expression of this protein is sufficient to explain the statisticalproperties of the cell-division time and the size added from cell birth to division. Keymodel insights are as follows:

• Distribution of the volume added from birth to division is independent of thenewborn cell volume, a hallmark of the adder principle (Fig. 3.3).

• The distributions of key quantities such as the added volume, division time,volume at birth and division are scale invariant.

• The noise in cell-division time increases with increasing newborn cell size (Fig. 3.2).

An important point to note is that if variation in ∆V is indeed a result of noisy

gene expression, then ∆V for successive cell-cycles should be independent. Indeed,

data shows a weak correlation between the volume added for mother and daughter

cells [7,8]. This result also argues that extrinsic fluctuations in parameters that exhibit

strong memory between mother and daughter cells cannot account for the statistical

fluctuations in ∆V .

35

Chapter 4

PART 2: OPTIMALITY IN HOST-VIRUS SYSTEMS

Life traits of virus are strikingly variable, ranging from highly infectious and

virulent to less virulent and chronic. Unveiling the mechanisms behind these different

viral strategies of host explotation remains a key challenge in biology.

The classic theory of parasite evolution shows that nature will select the virus

that maximizes the basic reproductive ratio (R0). This quantity represents the num-

ber of secondary infections resulting from one infected host. We can compute it by

understanding the dynamics behind host-virus interactions.

Let T be the target cell population available in the environment. Assuming this

population is near steady state, its dynamics can be described by the ODE

T = λ− dTT

. Under unconstrained conditions, if a virus V and a target cell meet, the former will

infect the later. Infected cells (I) will actively produce viruses until its death. This

phenomena is modeled as

T = λ− dTT − βTV (4.1)

I = βTV − dII (4.2)

V = bI − dV V, (4.3)

where β is the adsorption rate, b is the number of virus produced by an infected cell.

β and b might be interpreted as the infectivity of the virus. The death rates dI and dV

represent the death rate of the infected cell and virus, respectively. dI can be referred

T = λ/dC , I = 0, V = 0.

36

Our system will leave this state if and only if

R0 =λβ b

dC dI dV> 1, (4.4)

i.e., if the number of secundary infections is greater than 1.

Clearly, virus infectivity (β, b) acts to increase R0, whereas virulence (dI) de-

creases it by reducing the infection period. With this in mind, if the objective is to

maximize R0 (number of secondary infections), virus might attemp to evolve to infinte

infectivity (β, b) and zero virulence (dI) as suggested by Equation (4.4). Experimental

work had shown contradicting results, supporting the existence of both moderated in-

fectivity and virulence. Through the following chapters we propose scenarios in which

moderated infectivity and virulence might be advantageous to the virus through several

fitness definitions including the above discussed.

There is evidence suggesting that viruses with intermediate (moderated) adsorp-

tion rate are more adequate under structured host populations where virus-host local

interactions cannot be discarded. For instance, the local interaction between debris of

the lysed cell and newborn virus has been suggested as explanation for intermediate

evolved traits. Chapter 2 introduces a model that includes local interactions between

bacterial cell debris and newborn viruses (bacteriophage). We found that these inter-

actions may in fact explain the observed intermediate adsorption rate. We show this

by exploring single and competition dynamics of different virus strains.

How this moderated infection mechanisms are implemented in more complex

setups like human-virus interactions? In chapter 3 we study the infection by Human

Imunnodeficiency Virus. It has been shown that the infection of CD4+ T Cells by

HIV happens by two distinct mechanisms: cell-free transmission by free viruses (large

infectivity), and cell-cell transmission (intermediate infectivity) in which viral parti-

cles are transmitted directly across a tight junction or synapse between an infected

and an uninfected cell. We show that synapse-forming HIV (moderated) provides an

evolutionary advantage relative to non synapse-forming virus given a specific scenario.

37

Is moderate virulence of current virus an evolutionary advantage? Chapter 5

explores the way in which virus evolved to moderate temperateness in terms of the

propensity of a virus to enter lysogeny. Assuming scenarios with recurrent cycles of

good and bad conditions, we studied how multiplicity of infection (MOI) - the ability

of a virus to infect an already infected cell - drives the stochastic decisions of entering

or not into lysogenic mode. We found that temperate virus might use the lysogenic

path to protect themselves from extended periods of detrimental conditions.

In HIV infections, populations of quiescently infected cells form a ”latent pool”

(moderated virulence) which causes rapid recurrence of viremia whenever antiviral

treatment is interrupted. A ”cure” for HIV will require a method by which this latent

pool may be eradicated, by increasing the virulence of the virus. Current efforts are

focused on the development of drugs that force the quiescent cells to become active.

Previous research has shown that cell-fate decisions leading to latency are heavily

influenced by the concentration of the viral protein Tat. In chapter 6, we explore

therapeutic alternatives using combination of traditional and Tat based drugs.

38

Chapter 5

OPTIMAL ADSORPTION RATE: IMPLICATIONS OF THESHIELDING EFFECT

Since the evolution of parasites and pathogens is important to human [119], agri-

cultural, and wildlife systems, there is a mature theory that focuses on how infection

mechanisms may evolve. Given that viruses are the most abundant and simple entities

on the planet, they are frequently used as models for studying parasite-pathogen evo-

lution. In particular, parameters like replication, lysis time, adsorption among others

have been suggested as possible knobs used by viruses to drive infection [120–123].

Furthermore, adsorption rate has been proposed as one factor that the viruses may

easily tweak to gain maximum advantage from bacterial populations [124–126].

The classic theory of parasite evolution states that natural selection will max-

imize the number of secondary infections resulting from infecting a susceptible host

[127]. One way of doing so is by evolving the adsorption rate, which is the probability

of a virus infecting a bacteria upon direct contact. Under unconstrained environments,

the classical theory will predict that the virus will evolve to maximum (infinite) ad-

sorption rate. However, experiments show the unexpected emergence of virus with

moderate or intermediate adsorption rates [125,128]. How and under which conditions

this intermediate rate evolves remains to be poorly understood.

From the biological standpoint, several theories has been proposed. The occur-

rence of intermediate adsorption rates may be explained by the presence of host spatial

structure [129]. This spatial structure have importance to the local virus-host interac-

tions that can constrain the viral evolution into higher adsorption rates. Presumably,

the viral strains with high absorption rates tend to create a shielding effect in which

the local availability of healthy bacteria is reduced, producing more virus-infected cell

39

interactions, which lead to zero new viruses [127]. In other words, high adsorption rate

can be a curse for viruses since it can lead to premature discard if attached to debris

produced by cell lysis [123].

Spatial structure in the susceptible bacteria has been modeled using determin-

istic [130–132], and stochastic approaches [133–137]. Specifically, these models address

the problem of virus-bacteria coexistence rather than explaining why intermediate ad-

sorption rate may arise. To the best of our knowledge, there is only a recent work which

studies optimality on adsorption rate using agent-based modeling and modeling explic-

itly bacterial structure [123]. Specific debris modeling has been also used for studying

host-pathogen coexistence without studying theoretical aspects of the occurrence of

optimal adsorption rate under shielding effects conditions [138,139].

In this chapter, we propose an approach for modeling local interaction effects

through the inclusion of debris as a form of shielding. We use ODE tools to study the

impact of debris on the virus fitness. Specifically, we assume that newborn viruses are

locally surrounded by debris from the lysed cell. The effective burst size of an infected

cell is related with the propensity of a newborn virus to attach to lysed cell material. If

virus escapes from this local interaction it will find susceptible bacteria and create new

progeny. Our simple yet effective model shows the existence of optimal virus fitness in

single and competition assays that cannot be explained by classic parasite-host models.

This chapter is organized as follows: first classic virus-bacteria interaction mod-

els are introduced, and its implications for viral fitness are discussed. Then we describe

how local interactions can be included into the classic model, i.e., how debris is included

in the model. Later, we discuss single virus dynamics implications. Furthermore, com-

petition between strains is explored. Finally we present conclusions and further work.

5.1 Traditional virus dynamics model

We consider first a basic model for virus dynamcis introduced by [140]. Let C,

I, and V be respectively the number of healthy, infected bacteria, and virus particles.

40

Figure 5.1: Dynamics of the shielding effect. Virus attach to a healthy bacteria,injects its genetic material, infected cells produce new viruses, and thenthe infected bacteria lysis producing debris and newborn viruses. Thenewborn viruses can either attach to the debris (shielding effect) or escapefrom it and infect new susceptible bacteria.

The dynamics of these species is given by the ODE system

C =λ− dCC − rV C (5.1)

I =rV C − dII (5.2)

V =BdII − rV C − dV V. (5.3)

Healthy bacteria reproduces at a rate λ and dies at rate dC . Virus attacks bacteria at

rate rV C, where r is the adsorption rate. Once the infected bacteria dies (with latent

period 1/dI), it releases a burst of viruses of size B. Free virus may die at the rate dV .

It is easy to check that the steady state virus level of system (2.1-2.3) is

limt→∞

V =λr (B − 1)− dCdV

dV r. (5.4)

Eq. (2.4) can be seen as one way to measure virus fitness. Note that, for the virus to

obtain maximum fitness, adsorption rate should be infinite ( maximum virus load at

steady state is λ (B − 1) /dV ).

Alternatively, the baseline reproduction ratio may be used to represent the fit-

ness of the virus. Recall that the uninfected steady state C = λ/dC , I = 0, and V = 0

is unstable point (meaning the infection will take place) if

R0 =Bλr

dCdV> 1, (5.5)

41

where R0 is the baseline reproduction ratio and can be interpreted as the number of

viruses produced per infection. R0 can be used to infer the evolutionary outcome of

the system (2.1-2.3). For instance, we can infer from (2.5) that the virus should evolve

towards infinite adsorption rates in order to get the maximum fitness.

5.2 Modeling the shielding effect

Consider the basic lytic cycle of a virus depicted in Fig 2.1. A given virus attach

to a healthy bacteria, and injects its genetic material. Then, infected cells produce new

viruses during the latent period. The lysis of the infected bacteria releases debris and

newborn viruses. The newborn viruses can either attach to the debris (shielding effect)

or escape from it and infect new susceptible bacteria.

To model the shielding effect, we included two new species into the system

(2.1-2.3). First, we assumed that, once the virus is released after infected cell lysis

(Vs), this virus will be surrounded by cell debris (X). Cell lysis will release B virus

copies and q debris. The newborn virus will either attach to the debris and become

inactive, or escape at rate df and be able to reach suceptible bacteria. We assume that

after escaping, the shielding effect disappears due to fast debris dynamics. The virus

dynamics of this new system can be written as

C = λ− dCC − rV C (5.6)

I = rV C − dII (5.7)

Vs = bdII − rXVs − dV Vs − dfVs (5.8)

X = qdII − dXX − rXVs (5.9)

V = dfVs − rV C − dV V, (5.10)

where dX is the debris degradation rate. Assuming that the dynamics of the debris

(X) and the shielded virus (Vs) are assumed to be faster than free virus, susceptible

42

bacteria, and infected cell, this is

bdII − rXVs − dV Vs − dfVs =0 (5.11)

qdII − dXX − rXVs =0, (5.12)

the shielded virus Vs and the debris levels are given by

Vs =bdII

(df + dV ) + rX, X =

qdII

dX. (5.13)

Replacing Vs with Vs in (2.10), and recalling that we assumed faster dynamics for X

and Vs interactions, the burst size B can be rewritten in terms of the adsorption rate

(r) as

B (r) = bdf

df + rX. (5.14)

where b is the maximum burst size that can be obtain from infected cells. Now we can

replace the constant burst size in system (2.1-2.3) by the adsorption-dependent burst

size (2.14).

The burst size function (2.14) can be seen as the effect of the spatial structure

in the system (2.1-2.3). For instance, small escape rates of the virus (df ) can be

interpreted as debris clusters formed after cell lysis that protects susceptible bacteria

from being infected by the newborn viruses. Larger escape rates imply no debris

surrounding new viruses, allowing newborn viruses to attack bacteria once cell lysis

occurs.

5.3 Single virus dynamics

The adsorption-dependent burst size given by (2.14) provides new properties to

system (2.1-2.3). In particular, the maximum steady state virus load is only possible

if the adsorption rate is given by

r∗ =df√dCdV√

bλdfX − X√dCdV

. (5.15)

Fig. 2.2 shows the effect of constant and adsorption-dependent burst size in the steady

state virus load. Instead of infinite adsorption rate, optimal virus load requires a

43

finite and intermediate adsorption to avoid virus waste produced by the shielding ef-

fect. Moreover, note that the fitness defined by R0 becomes finite when adsorption-

dependent burst size is used, given by λbdf/(dCdV X).

Figure 5.2: Effect of constant and adsorption-dependent burst size in the steady statevirus load. Instead of infinite adsorption rate, optimal virus load requiresa finite and intermediate adsorption to avoid virus waste produced by theshielding effect.

Although single virus dynamics show emerging properties promoted by the

shielding effect, it is not clear how these properties will promote the evolution of the

virus to moderate adsorption rates. Next we will study the implications of having

adsorption-dependent burst size in competition between different viral strains.

5.4 Competition dynamics

To study competition between different strains we built a two-strain ODE sys-

tem by extending (2.1-2.3). We assume that there is an established virus V (which is

at steady state). A different virus strain Vm (invader) is introduced to the environment

44

at small copy number. The dynamics of the expanded system is given by

C =λ− dCC − rV C − rmCVm (5.16)

I =rV C − dII (5.17)

V =B (r) dII − rV C − dV V (5.18)

Im =rmCVm − dIIm (5.19)

Vm =B (rm) dIIm − rmCVm − dV Vm, (5.20)

where Im, and rm are the infected cell level and the adsorption rate of the new viral

strain introduced into the environment. Next, we derive the condition for the mutant

virus Vm to invade the established virus V .

Consider the dynamics (described in Eq. (2.16-2.20)) of a resident and an in-

vader virus with adsorption rates r and rm, respectively. Additionally, let the resident

virus be at its steady state level, i.e.,

C =dV

(B (r)− 1)r(5.21)

I =dCdV + λr − λrB (r)

dIr −B (r) dIr(5.22)

V =(B (r)− 1)λ

dV− dC

r(5.23)

We are interested in the local stability of the steady state solution C = C, I = I,

V = V , Vm = 0, and Im = 0. To this end, we compute the Jacobian

JC,I,V ,0,0 =

J 0

0 Jm

, (5.24)

where J is the matrix containing the partial derivatives of the dynamics of the resident

virus, and

Jm =

− dV rm(B(r)−1)r

− dV B (rm) dIdV rm

(B(r)−1)r−dI

(5.25)

contains the partial derivatives of the mutant dynamics. Each element of Jm represents

an increase or decrease in the number of infections and viruses. For instance, viruses

45

and infected cells die by degradation (dV ) and lysis events (dI), respectively. Alterna-

tively, the number of viruses increase by B (rm) dI . The expression dV rm(B(r)−1)r

represents

the number of mutant viruses required to produce new infections in presence of the

established strain.

Note that the stability of the system is dictated by the eigenvalues of the matrix

Jm. We use a simple yet powerful approach to study this eigenvalues. As proposed

by [141], we decompose Jm into the matrices

F =

0 B (rm) dI

0 0

, (5.26)

V =

dV rm(B(r)−1)r

+ dV 0

− dV rm(B(r)−1)r

dI

. (5.27)

F represents the alternatives available to produce a mutant virus. The processes in

which a mutant virus die are included in V . Given the matrices F and V , the stability

of the aforementioned system is dictated by the following theorem.

Next-generation theorem [141]: The maximum real part of all eigenvalues of Jm =

F − V is greater than 0 if and only if

ρ(FV −1

)> 1, (5.28)

where ρ (FV −1) is the maximum absolute value of all eigenvalues of matrix FV −1 �

Applying this theorem to eq. (2.26), it is straightforward to show that

ρ(FV −1

)=

B (rm) rm(B (r)− 1) r + rm

. (5.29)

Therefore, the mutant virus will invade if and only if

B (rm) rm(B (r)− 1)r + rm

> 1. (5.30)

Note that, when B (r) = B (rm) = b, Eq. (30) reduces to

b rm(b− 1)r + rm

. (5.31)

46

In this extreme case the mutant virus will invade when rm > r, i.e, a virus should

evolve to its maximum adsorption rate.

The above result can be easily extended to study burst size in the form given by

Eq. (2.14). Let the burst size be defined by (2.14) and assume for simplicity that the

debris X is constant. Then the optimal mutant adsorption rate required for invasion

is given by

r∗m =

√df r(b df − df − r X)

dfX + rX2. (5.32)

Note that r∗m increases as the maximum burst size b becomes larger. This can be

explained intuitively as follows. A larger burst size implies that there are more newborn

viruses escaping from debris. Once virus escapes, it is better to have a larger adsorption

rate (rm) that allows infecting more bacteria.

Eq. (2.32) also shows an inverse relationship between the optimal adsorption

rate (r∗m) and the amount of debris (X). Virus might evolve to mutants with moderate

(intermediate) adsorption rates when the bacterial debris (X) in the vicinity of the

newborn viruses increases. A moderate strain will have a lower chance of attaching to

bacterial debris, and hence a higher chance of escaping from the debris and reaching

more healthy bacteria.

We studied the system (2.16-2.20) with shielding effect (adsorption-dependent)

and without it (adsorption-independent) burst size. In the adsorption-dependent case,

we set the adsorption rate of the established virus (r) to an intermediate value. For

the invader virus we studied two cases, one where its adsorption rate is lower than the

established virus adsorption rate (rm < r), and the other when the invader adsorption

rate is larger (r < rm). Results are shown in Fig. 2.3. When the shielding effect

is included in the system, any mutant adsorption rate (rm) larger or smaller than the

established virus (r) will not invade the system. In contrast, in the absence of shielding

effect, if the invader has a larger adsorption, invasion will take place as expected.

The classic theory of parasite evolution states that viruses will evolve to infinite

adsorption rates in order to attain its maximum reproductive ratio R0. However,

47

Figure 5.3: When the shielding effect is included in the system, any mutant adsorp-tion rate (rm) larger or smaller than the established virus (r) will notinvade the system. In contrast, in the absence of shielding effect, if theinvader has a larger adsorption, invasion will take place as expected. Pa-rameters used were: b = 100, dC = 0.1, dV = 0.001, λ = 100, df = 0.01,X = 1000, r = 10−4, dI = 1, lower adsorption rate rm ∈ {10−6, 10−3}.

experiments shows that this may be not the case, and intermediate adsorption rates are

the natural selection in evolution. Moderate adsorption rate might arise in situations

where healthy bacteria is unavailable or have poor quality as proposed by [142]. Despite

the body of research that attempts to explain why moderate adsorption rates exists,

current models are unable to explain the existence of optimal adsorption rate under

competition situations.

This chapter presents the first attempt to explain why virus may evolve into

moderate instead of infinity adsorption rates. Our approach uses classic ODE analysis

48

to study the effect of cell debris (shielding effect) on virus fitness under both, individual

and competition scenarios. We included this shielding effect by assuming that the burst

size is a decreasing function of the adsorption rate. This assumption is valid as long as

the interaction between the newborn virus and the cell debris produced is fast enough to

produce a net decrease in the viral burst size. We found that this adsorption-dependent

burst size produces a finite basic reproductive ratio, enables moderate adsorption rate

virus to maximize the steady state virus load, and allows establishment of viruses with

moderate adsorption rates when competing with larger adsorption rate strains. This

result expose the importance of virus-bacteria local interactions on the study of traits

evolution.

Until now we explored a burst size which is a decreasing function of the adsorp-

tion rate. One question raises after this study: which class(es) of burst size functions

warranties optimal adsorption rates in both individual and competition scenarios. Ad-

ditionally, we considered a deterministic adsorption rate, however, adsorption rate may

be a distribution (as shown in experiments [143, 144]) in which natural selection will

choose moderate adsorption instances. How this selection arises is also an intriguing

open question. Furthermore, debris is not the only mechanism that may produce the

shielding effect. Other alternatives include multiplicity of infection [145], presence of

resistant bacteria, or high density of infected cells in the neighborhood of the newborn

virus.

In this chapter we modeled the lysis process using rate equations. In reality,

lysis is an exact delayed process. Moreover, optimal lysis time is required to ensure

virus establishment [146]. Future work includes studying the role of both shielding

effect and lysis time on viral fitness.

49

Chapter 6

THE EFFECT OF MULTIPLICITY OF INFECTION ON THETEMPERATENESS OF A BACTERIOPHAGE: IMPLICATIONS FOR

VIRAL FITNESS

Since the evolution of parasites and pathogens is important to human [119], agri-

cultural, and wildlife systems, there is a mature theory that focuses on how infection

mechanisms may evolve. Given that viruses are the most abundant and simple enti-

ties on the planet, they are frequently used as models for studying parasite-pathogen

evolution. In particular, parameters like replication, lysis time, and adsorption among

others have been suggested as processes used by viruses to drive infection [120–123].

Bacteriophages are viruses that infect bacteria. Their natural environment is

challenging, characterized by fluctuating host cell populations and other sources of

stress to the phage [147]. In this situation, a phage has two courses of reproductive

action: lysogenization or initiation of the lytic cycle (see Fig. 1). Lysogenization is a

means for the phage to lie dormant inside of a bacterial host by integrating viral nucleic

acid into the genome of the host cell [148, 149]. A phage thus integrated is called a

prophage. While the prophage remains latent, it does not impede the host cell in any

way. The bacterium will continue thriving and propagating, copying and transmitting

the prophage into its progeny. This allows the phage to reproduce without exposing

itself to the detrimental effects of the outside environment. The lysogen has the ability

to maintain its current state of latency or undergo induction. When induction occurs,

prophage DNA is cut off from the bacterial genome and coat proteins are produced via

transcription and translation of the phage DNA for the regulation of lytic growth.

In the lytic cycle, the genome of the phage is inserted into the cytoplasm of the

bacterium. The DNA resides separately from that of the genetic material of the host.

50

Replication of the phage begins and, once many phage components have been created,

new phage are produced. Over time, the phage will begin to accumulate within the host

cell. This eventually results in the lysing of the bacterium and the release of the free

phages. A phage that has the ability to enter lysogeny is called a temperate phage, and

its temperateness can be defined as the propensity of the phage to enter a lysogeny. The

idea of temperateness has been greatly deliberated and analyzed [150, 151]. However,

the main questions still stands: what advantage does a phage derive by being able to

switch between the lysogenic cycle and the lytic cycle?

Virus

Bacteria

Viral genome

Lysogen

Host genome

Integratedviral genome

Lytic

Figure 6.1: Plasticity of the temperate phage. Under the lysogenic cycle, the cell caneither undergo the lytic cycle with a probability of 1−φ or integrate theprophage into the genome of the host cell with a probability of φ.

One mechanism phages might use to regulate temperateness is by means of the

multiplicity of infection (MOI) – the ability of a phage to infect an already infected

cell. To gain a more concrete understanding of the behavior of the temperate phage, we

analyzed how MOI may affect the probability of entering the lysogenic cycle. Math-

ematical modeling of a multitude of fluctuating environments [130–137] allow us to

theoretically and quantitatively understand the adaptive nature of fixed and plastic

latency.

In this chapter, we explore the advantages of choosing the lysogenic path under

51

fluctuating environments. Additionally, we explore the additional advantages provided

by MOI. The chapter is organized as follows: first we describe the modeling approach

used to represent the dynamics of the different species under fluctuating environments.

Then we explore the lysogeny-lytic decision made by the phages. We study the impact

of MOI on the overall fitness of the phage and present some conclusions.

6.1 Model

We modeled the fluctuating environments using a similar approach as in [142].

Consider a situation where the environment fluctuates between ”good” and ”bad”

conditions. The ”good” conditions allow for an environment in which the phage is

proliferating efficiently through a replenishing population of bacteria. The ”bad” con-

ditions result in the entirety of the free phage population decreasing to 0. This would

be consistent with an environment of extreme host scarcity. The phage has no control

over its environment. However, it does have the ability to infect a host with a probabil-

ity k > 0 assuming the ”good” conditions. Then, with a probability of φ, the lysogenic

cycle can be induced. This allows the phage to reproduce after a period of dormancy

by incorporating it’s own DNA into the genome of the host. Furthermore, the phage

has a probability of (1 − φ) to enter the lytic cycle. This causes the bacteria to lyse

and, in turn, create new free phage. These new phage are represented by BV where

B is the phage burst size per infection. Fluctuations in the free phage and lysogen

populations during the ”good” condition are represented by

V(1-φ) k−−−→ B V, V

φk−−→ L, (6.1)

During the ”bad” condition, the free phage die out completely and the lysogen

population decreases at a fixed rate. The change of lysogen population under ”bad”

conditions is given by

Lα−−→ ∅, (6.2)

where α > 0 is the degradation rate of the lysogens. Once the ”bad” condition ends,

all lysogens release their phage (BL), and a new ”good” condition starts. This does

52

not imply the existence of a sensor in the lysogens that recognize the good condition.

This is a simplification, without loss of generality, since the contribution of new free

phage will be proportional to the lysogen population at the beginning of the good

time. Additionally, the exponential growth via the lytic pathway rapidly overwhelms

any residual contribution of remaining lysogens from the previous round.

The dynamics of such phenomena can be represented by the hybrid system

depicted in Fig. 5.2a. The ellipses represent the good and bad environment situations.

Arrows represent birth, death or environmental switching events. Free phage is created

in bursts of fixed size B at a rate (1− φ)k v, where v represents the current number of

phage within the system. Alternatively, new infections might choose the lysogenization

path, which happens at a rate φ k v, adding an extra lysogen to the system. A timer

variable τ keeps track of the elapsed time in the good condition until time τg is reached.

The rate in which it takes to cycle through one iteration of good conditions is given by

δ(τ − τg). After this time, the environment switches to bad conditions, resetting the

free phage population and the timer τ to zero. In bad conditions, death solely occurs

at a rate αl. Once (τ = τb), the environment returns to good conditions,τ is reset

to zero, and initial conditions are set to a free phage population proportional to the

lysogen population at the end of the bad time, with lysogen population set to zero.

6.2 Why do Bacteriophages display Temperateness?

To study the above hybrid system, an equivalent model can be constructed by

assuming the deterministic counterpart (see Fig. 5.2b). Let τg and τb be the time spent

in the good and bad environment, respectively. Starting with a single copy, the free

phage and lysogen count at the end of the good environment are given by

v (τg) = ekB(1−φ)τg , l (τg) =φ(ekB(1−φ)τg − 1

)B(1− φ)

. (6.3)

Note, that when the probability φ of becoming a lysogen is 0, the lysogen count is

zero. For example, a completely lytic virus will produce zero lysogens. Although, if

(φ = 1), the average lysogen count is k τg. Fig. 5.3 shows the lysogen population at

53

a) b)

Figure 6.2: Hybrid systems describing the fluctuating environment that lysogeniccapable phage might face. a) The environment dynamics is describedby one birth-death processes and a pure death process. Under goodenvironment situations (g = 1), phage might produce bursts of B freephage particles at a rate (1 − φ) k v. Alternatively, the virus might golysogenic. A timer τ keeps track of the elapsed time in each condition.Once time spend under good conditions is τg, the environment switchesto bad conditions by resetting τ , g, and v to zero. Under bad conditions,lysogenic cells die at a rate αl. When the time spent in the bad conditionsis complete (τ = τb), the environment switches back to good conditionsby resetting g = 1, v = Bl, l = 0, and τ = 0. b) Deterministic version ofthe hybrid system on part a).

the probability range of φ ∈ [0, 1]. Note that phage fitness is optimized for a value

between 0 and 100% chance of lysogeny.

Next, we speculate how a phage can spread over multiple rounds of good-bad

conditions. Since the free phage is wiped out completely during bad conditions, the

only way for the phage to propagate is through the lysogenic cell. Therefore, we are

interested in the conditions that allow for the lysogenic cells to thrive. At the end of

the first round, the lysogenic count is given by the equation 6.3

l0 =φ(ekB(1−φ)τg − 1

)B(1− φ)

e−ατb . (6.4)

54

Subsequent rounds (n > 1) are described by

ln =φ(ekB(1−φ)τg − 1

)(1− φ)

ln−1e−ατb (6.5)

=1

B

(φ(ekB(1−φ)τg − 1

)(1− φ)

e−ατb

)n

. (6.6)

The lysogenic population will grow unbounded when

φ(ekB(1−φ)τg − 1

)(1− φ)

> eατb , (6.7)

i.e., when the amount of phage (in form of lysogens) per infection during the good

condition is larger than the average lost per infection during the bad condition.

Now, the question is how this optimal lysogen count reflects on the phage pop-

ulation in subsequent rounds of good-bad environments.

k = 0.10k = 0.15k = 0.20

Probability of becoming lysogen

Lyso

ge

n c

ou

nt

(avg

)

Reproduction rate

Viral reproduction rate

Op

tima

l lyso

ge

nic

pro

ba

bili

ty

Op

tima

l lyso

ge

nic

po

pu

latio

n

Figure 6.3: Effects of lysogenic probability on phage infection. a) Population countvs lysogenic probability for several infection rates. Note that averagelysogenic count is optimized for a value between 0 and 100% chanceof lysogeny. b) The optimal chance of lysogeny decreases as the phagereproduction rate increases.

55

Since the free phage count is zero during the bad conditions, the lysogen con-

centration at the end of this cycle is given by

l (τg + τb) =φ(ekB(1−φ)τg − 1

)B(1− φ)

e−ατb . (6.8)

At the end of the next good condition, the free phage concentration can be written as

v (τg + τb + τg) =φ(ekB(1−φ)τg − 1

)(1− φ)

e−ατbekB(1−φ)τg . (6.9)

Fig. 5.4 shows the free phage concentration profile for the nominal parameter values.

Note that, similar to the lysogen count, intermediate lysogen probabilities produce

maximum free phage counts.

Vir

us

cou

nt (a

vg)

Probability of becoming lysogen

Figure 6.4: Free phage population count vs chance of lysogeny. Population count isoptimized at a value between 0 and 100 % chance of lysogeny.

6.3 Probability of survival of a lysogen

Is there an optimal lysogen count that characterizes the survivability of the lyso-

gen cells after the end of a good-bad environment sequence? To answer this question

we model the dynamics of free phage and lysogens as deterministic. Under bad condi-

tions, we assumed that lysogens dynamics obey a pure death process. Let the hybrid

system in Fig. 5a represent the dynamics of the free phage and lysogen species at the

end of the bad conditions. Since the dynamics of the lysogen species during the bad

56

conditions results in the degradation of the lysogen count, the probability of having x

lysogens can be written as

P (x, τb) =

x0

x

e−ατbx(1− e−ατb

)x0−x , (6.10)

where x0 is the lysogen count at the end of the good condition. The probability that

at least one lysogen survives the bad conditions is then

Ps = 1− P (0, τb) . (6.11)

The term h = 1 − e−ατb in equation (6.10) represents the probability of extinction

for one single lysogen during bad conditions. This probability is dictated by the ratio

between the average lifetime of the lysogen and the length of the bad environment.

The larger the ratio between these two, the larger the probability of extinction for one

single copy.

From equation (6.10) when the probability of extinction is close to 0, the survival

of one single lysogen will suffice to preserve this virus across multiple rounds of good

and bad conditions. For extinction probabilities close to 1, virus should compensate

by getting the maximum profit out of the good conditions, which is only achieved

at moderated lysogenic probabilities. Note that when viral reproduction rate is large

enough to produce lysogen counts >> 1, the probability that at least one lysogen

survives the bad conditions is 1 for φ < 1. Fig. 6.5b shows the probability distribution

Ps.

6.4 The Effect of Multiplicity of infection (MOI)

Consider the expanded version of the model in Fig. 5.2b where the dynamics of

the lytic cell are slow. We represent the transient dynamics by the set of ODEs

l =φ k v + φ a i v (6.12)

i =(1− φ) k v − dii− φ a i v (6.13)

v =B di i (6.14)

57

Lyso

ge

n s

urv

iva

l

pro

ba

bili

ty

Probability of becoming lysogen

0.997

0.998

0.999

Single lysogen

extinction probability (h)

Figure 6.5: Hybrid system describing the switching dynamics of the good and badenvironment. a) Population dynamics under good conditions are mod-eled as a deterministic system of ODEs. Population dynamics under badconditions are modeled as a pure death process. b) Survival probabilityof lysogenic cells at the end of the bad conditions. Note that this prob-ability is optimal when the phage is tempered. Additionally, the largerthe extinction probability of a single lysogen, the lower the survival prob-ability.

where i describes the rate at which the infection undergoes the lytic pathway. The

death rate of the lytic cells is given by di. Parameter a determines the adsorption rate

of the phage to the already infected cells. These cells might become lysogens with the

same probability as the new infections (φ). Note that the phage reproduction rate k

is defined in terms of a, i.e., k = ac where c is a fixed number of healthy bacteria cells.

We explored situations with and without MOI effects. To this end, when the

phage is unable to re-infect an already infected lysogen, the term φ a i v disappears from

equations (5.12) and (5.13). For MOI-enabled phage we varied the adsorption rate (a)

58

and kept constant the phage reproduction rate (k). Since k = a c, an increase in phage

adsorption implies a reduction in the constant number of bacteria in the environment.

Fig. 5.6 shows the effects of MOI for different adsorption rates. The blue solid

curve represents the non-MOI version of each phage adsorption rate. Note that by

adding MOI, the survival probability increases. Additionally, the optimal lysogeny

probability decreases. As the phage becomes aggressive, the optimal lysogenic proba-

bility reduces and the optimal survival probability increases.

a = 10-4a = 10-3

w/o MOI

Adsorption rate

Probability of becoming lysogen

Pro

ba

bili

ty o

f ly

so

ge

n

su

rviv

al

Figure 6.6: Effects of the multiplicity of infection (MOI) on the survival probabil-ity. The optimal chance of lysogeny decreases for MOI enabled phagescompared with phages without MOI. The blue solid curve represents thenon-MOI version of both MOI enabled phage (a = 10−4,a = 10−3)

When the availability of the bacterial population is held constant, classical the-

ory of parasite evolution states that aggressive phage strains will be selected for in

their environment. In this case, the probability of becoming lysogenic should be close

to zero. However, the role of lysogeny on viral fitness under these conditions has yet

to be fully uncovered. In this chapter, we observed a series of dynamic environments

with constantly changing conditions. These conditions oscillate between those where

the bacterial populations is readily available to the phage, followed by periods where

there is no bacterial population to infect. During such periods, the free phage has a

null chance of survival.

59

We found that under such scenarios, lysogenic cells play a key role in the preser-

vation of the phage for subsequent generations. Moreover, only temperate phages

(those with intermediate lysogenic probabilities) maximize phage population in good

conditions, lysogenic population at the end of bad conditions, and virus survival prob-

ability. How temperate the phage should be depends on the phage reproduction rate,

which depends on how aggressive the phage is (adsorption rate). The more aggressive

the phage is, the lower the required probability of becoming lysogenic and therefore,

the larger the optimal survival probability.

Using this framework, we also studied the impact of the multiplicity of infection

mechanism implemented by some phage species. We found that in some cases, MOI

might double the probability of lysogen survival, and reduce the optimal probability

of becoming a lysogen (φ) by a factor of 10.

60

Chapter 7

CONDITIONS FOR INVASION OF SYNAPSE-FORMING HIVVARIANTS.

Human Immunodeficiency Virus (HIV) is a human retrovirus that infects certain

immune cells, primarily the CD4+ helper T cells and the macrophages. Untreated

infection can lead to the eventual collapse of the immune system, resulting in severe

immunodeficiency and death due to opportunistic infections [152]. Over 34 million

people are infected, with as many as 2.5 million new infections each year [153].

The lifecycle of the HIV virus has been extensively studied. The virus particle

consists of two positive single-stranded RNA copies of the viral genome, together with

the functional HIV enzymes Vif, Vpr, Nef, Protease, Integrase, and Tat, are enclosed in

a protien capsid. This capsid is in turn encased within a lipid-bilayer envelope studded

with the viral glycoproteins gp41 and gp120.

The envelope protein gp120 has a high affinity for the cell marker CD4, which

is a characteristic marker of the helper-T cells and the macrophages [154], and to a

co-receptor, either the CCR5 or CXCR4 transmembrane proteins on the target cells

[155, 156]. Binding of gp120 to these two receptors results in a conformational change

which exposes an active site of the viral protein gp41. The exposed gp41 mediates fusion

of the viral membrane with the host cell membrane, which decapsulates the virus [157].

The viral membrane and all integrated proteins, including gp41 and gp120, becomes

part of the host cell’s membrane.

Once the virus has entered the host cell, viral reverse-transcriptase creates a

DNA copy of the viral RNA, a step that may be interrupted by the presence of reverse

transcriptase inhibitors. The viral DNA is transported to the cell nucleus, and inte-

grated into the host-cell chromosomes by the viral integrase enzyme. This step can

61

be interrupted by the presence of integrase inhibitors. The integrated HIV genome is

then expressed by the normal cellular RNA transcription machinery, although a hair-

pin structure in the evolving HIV RNA transcript can result in aborted transcription

unless the viral protein Tat is present. The viral RNA transcript is transported to the

cytosol, where ribosomes transcribe several non-functional super-proteins, which must

be cleaved by the viral protease enzyme into their functional forms. The virus products

aggregate on the cell surface and form viral particles.

These viral particles bud off of the surface of the host cell and are released into

the surrounding fluid, able to infect other cells. However, prior to the formation of

mature, budding HIV particles, there is an accumulation of gp41 and gp120 complexes

on the surface of the infected cell. In the same manner in which these viral proteins

mediate the binding of and fusion of viral envelopes to the target cell, they are also

capable of allowing the infected cell to bind to an uninfected target cell and partially

fuse membranes, resuling in the formation of a viral synapse. Formation of the synapse

allows for the direct transmission of viral particles between the infected and the unin-

fected cell [158–166]. This has been observed experimentally, and it has been shown

that as many as several hundred virus particles can be transmitted across a single

synapse [167].

It has been suggested that the cell-cell transmission pathway provides an evolu-

tionary advantage, either by allowing the virus to evade the host immune response [163],

to overwhelm antiviral drug activity [168, 169], or simply by a more efficient mode of

infection. Previous modeling work considering the potential benefits of cell-cell trans-

mission has shown that the optimal number of viruses transmitted by a synapse is

small [170, 171]. This previous work did not explicitly consider the competition be-

tween synapse-forming and non-synapse forming viruses, as we do here. The previous

work was also based on nominal parameters, where we use parameter values identified

from experimental patient data [172].

We present a novel model of HIV virus dynamics that accounts for transmission

by both the cell-free and cell-cell pathways. We analyze the stability of the stationary

62

points of this model, and derive bifurcation conditions. We show that the steady-state

viral loads in a synaptic transmission model increase as a function of the probability

of successful infection given a cell-entry event. We further show that the steady-state

virus load increases with synaptic multiplicity of infection when the fraction of total

burst size is small, but begins to decrease with increasing multiplicity of infection once

a relatively small fraction of the burst size is reached. We show that a synapse forming

virus variant will successfully invade against an established non synapse-forming virus

when the muliplicity of infection as a fraction of total burst size is less than a critical

value determined primarily by the fitness of the non synapse-forming virus.

The chapter is organized as follows. Section II introduces the basic model of

HIV dynamics as developed by Perelson et al. [173–175]. Section III introduces the

model of synapse-forming virus, develops stability conditions on the stationary points

of the model, and explores the effect of varying muliplicity of infection and probability

of successful infection on the steady-state virus loads. Section IV introduces a compe-

tition model between synapse-forming and non synapse-forming virus, and develops a

bifurcation condition for the stability of the stationary point with no synapse-forming

virus. Section V summarizes the results, and discusses implications for HIV treatment

and future work.

7.1 HIV Model

The free virus transmission mechanism is described using the extensively studied

model introduced by [173]. In this model the behavior of uninfected, infected cells and

63

Parameter Value Units Biological meaning Parameter Value Units Biological meaning

λ 7× 102 cellsµL×day Uninfected birth rate k 2× 103 copies

cell×day Copies of virus per cell

dT 0.1 1day Uninfected death rate βf 2× 10−6 mL

copies×day Rate of uninfected-virus interaction

dI 1 1day Infected cells death rate βs 10−5 µL

cells×day Rate of infected-uninfected interaction

dV 23 1day Virus death rate

Table 7.1: Parameter values for simulations on this chapter. All parameters valuesexcept βs were taken from [172]. Rate of infections by free virus pathway isassumed to be 20 times greater than rate of infections by synaptic pathway.

HIV virus is given by

T = λ︸︷︷︸T-cell

Production

− dT T︸︷︷︸T-cell

Death

− βf T Vf︸ ︷︷ ︸Free Virus

Infection

(7.1a)

If = βf T Vf︸ ︷︷ ︸Free Virus

Infection

− dI If︸︷︷︸Infected

Cell Death

(7.1b)

Vf = k If︸︷︷︸Free Virus

Production

− dV Vf︸ ︷︷ ︸Free Virus

Death

. (7.1c)

Here T (t), If (t) and Vf (t) represent uninfected, infected cells and virus population

at time t, respectively. The rate of production of uninfected cells is represented by λ.

Death rate of uninfected, infected cells and virus are dT , dI and dV . k represents the

number of free virus particles produced per infected cell per time unit. The infection

rate is given by βf . Table 8.1 shows these parameters and experimental values for them

obtained from [172].

If we assume that virus copies die at high rate greater than infected cells death

rate, this is dV >> dI , then virus population can be assumed in steady state, which

means Vf = kdVIf . Thus Equation (8.2) reduces to

T = λ− dT T −k

dVβf T If (7.2a)

If =k

dVβf T If − dI If . (7.2b)

64

In order to know if there is a successful infection in a person, steady state analysis of

Equation (7.2) is performed. Above Equation has two steady state solutions:

T =λ

dT, If = 0, (7.3)

which means there is not infection in the host, and

T =dTdVβfk

, If =βfkλ− dIdTdV

dIβfk, (7.4)

meaning virus infection succeed. Determining if local stability take place at point

represented by Equation (7.3) is equivalent to determine if infection will take place by

means of the free virus transmission mechanism. The eigenvalues at Equation (7.3)

are −dT which is always negative, and −dI +λkβfdV dT

which could be positive or negative.

Let

R0f =λkβfdV dIdT

(7.5)

be the basic reproductive ratio of HIV infection by means of the free virus pathway. If

R0 > 1 then Equation (7.3) is unstable and the infection will take place.

7.2 Modeling Synaptic Virus

Equation (8.2) describes transmission by free pathway. However that is not

the only way of HIV transmission. Also infection replication may happen by direct

interaction between cells, called synaptic transmission. When cells interact with others,

sometimes form channels known as synapses. When infected and uninfected cells form

synapses, virus copies coming from the former infect the later.

We modify Equation (8.2) in order to model synaptic transmission. Let Vs (t)

and Is (t) be population of synaptic virus and cells infected with synaptic virus at time

t, respectively. Also let s be the synaptic size, which is the amount of virus sent through

a given synapses. Besides free virus infection βf T Vs, we add an extra production of

infected cells given by p (s) βsTIs. Here βs is the rate of interaction between infected

and uninfected cells. Function p (s) is the probability that an uninfected cell will

65

get infected by receiving s virus particles through synapses, given there is interaction

between infected and uninfected cells. Probability p (s) is defined as

p (s) = f (s) σ (s) , (7.6)

where σ (s) is the probability that uninfected and infected cell form synapses given there

is interaction. f (s) is the probability that sending s viruses through given synapses

leads to an infection, and can be any monotonically increasing function on s. If we

assume the probability that s copies infect a cell as a binomial distribution, with each

virus copy having probability r of successful infection then

f (s) = (1− (1− r)s) , (7.7)

i.e. f (s) is the probability that at least one of s virus has successful infection given

there is synapses.

There are two possible scenarios for synapses formation: infected-uninfected

and infected-infected synapses. The former leads to an infection with probability p (s).

Therefore there is a reduction of s σ (s) βsT Is virus copies that cannot be used in further

infections. The other scenario arises because there is no discrimination mechanism

that leads infected cells to form synapses with uninfected cells only, thus infected-

infected interactions also should occur. Infected-infected synapses lead to a waste of

s σ (s) βsI2s virus copies that does not produce any additional infection, because both

cells are already infected. Figure 7.1 shows all three possible synaptic virus pathways:

free virus transmission, infected-uninfected and infected-infected virus transmission

through synapses.

Using the synaptic mechanism illustrated above and including it in (8.2) leads

66

(a) (b)

(c)

Figure 7.1: Synaptic virus mechanism. A synaptic virus has the capability of infectcells by means of free pathway (a) and also through synapses forma-tion (b). In the free pathway (a), infected cells produce RNA strings(red lines) using virus information stored in its genome (blue and redline), encapsules them (blue and red concentric circles) and send thiscapsids outside the cell. Uninfected cells absorbe them releasing RNAvirus strings (opened blue circle) which integrates with cell’s DNA (blueline). Synaptic interactions may occur between infected and uninfected(b) or infected-infected cells (c). The virus copies in (b) sent throughsynapses are not used in the infection of other cells.

to

T = λ− dT T − βf T Vs − p (s) βs T Is︸ ︷︷ ︸Synaptic

Infection

(7.8a)

Is = βf T Vs − dI Is + p (s) βs T Is︸ ︷︷ ︸Synaptic

Infection

(7.8b)

Vs = k Is − s σ (s) βs (T + Is) Is︸ ︷︷ ︸Reduction of

Virus Production

−dV Vs. (7.8c)

67

If we assume synaptic virus copies die at a greater rate than infected cells

(dV >> dI), then virus is in steady state (Vs =(1− s

kσ (s) βs (T + Is)

)kdVIs) and

Equation (7.8) reduces to

T = λ− dT T (7.9a)

−(

1− s

kσ (s) βs (T + Is)

) k

dVβfT Is

− p (s) βs T Is

Is =(

1− s

kσ (s) βs (T + Is)

) k

dVβfT Is − dI Is (7.9b)

+ p (s) βs T Is,

which have two stationary points, one of them being the uninfected state

T =λ

dT, Is = 0. (7.10)

Infection will occur (this point is unstable) if

R0s =

(1− s

kσ(s)βs

λ

dT

)R0f + p (s) βs

λ

dIdT> 1. (7.11)

The other stability point is not difficult to calculate, however is not included here due

space limits.

Figure 7.2 illustrates the effect of small fraction of virus particles sent through

synapses. Here B = ka

represents the burst size of the cell which is the number of

virus particles that an infected cell produces over its lifespan. sB

is the synaptic size

as a fraction of this burst size. For small values ( sB< 0.1) steady state level of virus

load increase more than 20%. Increasing the probability of infection of a single virus r

increases speed of infection growth. In the other hand Figure 7.3 shows that increasing

sB

close to 1 (s close to burst size) leads to a decreasing behavior of virus load. Table

8.1 shows parameter values used for all simulations on this chapter.

68

0.00 0.05 0.10 0.15 0.20 0.25 0.30r11 000

11 500

12 000

12 500

13 000

13 500

Virus Copies

Figure 7.2: Steady state behavior of virus load with different fractions of synapticsize s

B, with probability r of successful infection. Increasing virus load

when sB< 0.1 is observed. Fast growth is observed for r close to 1. Burst

size used on this simulations is B = 2 × 103. Probability of formingsynapses used is σ (s) = 1

0.02 0.04 0.06 0.08 0.10

s

B

11 000

11 500

12 000

12 500

13 000

13 500

Virus Copies

Figure 7.3: Steady state behavior of virus load for different synaptic size fractionssB

, with probability r = 0.1 of successful infection. Steady-state virusload decreases with s

Bwhen s

B> 0.1, and drops below the non synapse-

forming equilibrium for sB∼ 1. Burst size used on this simulations is

B = 2× 103. Probability of forming synapses is σ (s) = 1

7.3 Competition Model

In order to determine if synaptic virus outperform free virus mechanism, we

propose a new competition model between synaptic and non-synaptic virus. Merging

69

(8.2) and (7.8) into a new competitive system leads the new model

T = λ− dT T − βf T (Vs + Vf )− p (s) βs T Is (7.12a)

If = βf T Vf − dI If (7.12b)

Vf = k If − dV Vf (7.12c)

Is = βf T Vs − dI Is + p (s) βs T Is (7.12d)

Vs = k Is − s σ (s) βs (T + Is + If ) Is − dV Vs. (7.12e)

Note that in Equation (7.12a) βf T (Vs + Vf ) represents the total infection rate by the

free virus transmission for both non-synaptic and synaptic virus. In this model, cells

infected by synaptic virus (Is) are able to establish synaptic interaction with uninfected

(T ), cells infected by free (If ) and synaptic virus (Is).

In order to study stability of the competition model we assume dV >> dI , thus

Equation (7.12) reduce to

T = λ− dT T (7.13a)

− k

dVβf T

((1− s

kσ (s) βs (T + Is + If )

)Is + If

)− p (s) βs T Is

If =k

dVβf T If − dI If (7.13b)

Is =(

1− s

kσ (s) βs (T + Is + If )

) k

dVβfTIs − dI Is

+ p (s) βs T Is.

Determining if synaptic outperforms free mechanism is equivalent to find out if given

there is infection by means of the free virus, adding ε copies of synaptic virus implies

the extinction of the free virus. Thus, assuming that free virus is present and at steady

state (Equation (7.4)), the stability analysis of Equation (7.13) reduces to determining

70

Free Virus

outperforms

Synaptic Virus

outperforms

1 2 3 4 5 6 7 8 9 10R0 f

0.4

0.5

0.6

0.7

0.8

0.9

1.0

s

B

Figure 7.4: Bifurcation diagram showing regions where synaptic virus outperformsfree virus. The larger basic reproductive ratio of the free virus R0f is,the smaller the region where the synaptic virus outperforms gets.

whether

∂Is∂Is

=βsσ (s)

βfk2(dV dIkf (s)− dV d2

Is

+ dV dIdT s− 2k sIsβf )

− βsσ (s)

kλs (7.14)

is positive (unstable) for Is = 0. This is satisfied if

s

B<

f (s)

1− dT/dI +R0fdT. (7.15)

Figure 7.4 shows a bifurcation diagram that represents the region of values, for

synaptic size fraction sB

and basic reproductive ratio of free virus R0f , where synaptic

outperforms free virus. Note that the boundary in Equation (7.15) does not depend

on the reproductive ratio of synaptic virus R0s

Figure 7.5 shows competition simulations between synaptic and non-synaptic

virus.

In this chapter, we have introduced a novel model of HIV dynamics accounting

for both the cell-free and cell-cell mechanisms of viral transmission. We explicitly

consider the increase in likelihood of infection due to multiple cell entry events, the

reduced production of free virus by cells forming synapses, and the loss of virus due

71

300 400 500 600 700Days0

5000

10 000

15 000

20 000

25 000

30 000

Virus Copies

(a) R0f = 2, sB < 0.91

1000 1200 1400 1600Days0

10 000

20 000

30 000

40 000

Virus Copies

(b) R0f = 5, sB < 0.71

2000 2500 3000 3500Days0

10 000

20 000

30 000

40 000

50 000

Virus Copies

(c) R0f = 10, sB < 0.52

Figure 7.5: Dynamic behavior of competitive model built in this work. On thissimulations three values of basic reproductive ratio for the free virus R0f

are shown with their respective conditions defined by Equation (7.15).Population of free virus and infected cells with free virus are in steadystate (Equation (7.4)) for all three cases. There is no infected cells withsynaptic virus at time 0. One particle of synaptic virus is present att = 0. The fraction of synaptic size is s/B = 0.2. In all three casessynaptic virus outperforms free virus regardless increasing value of R0f .

to synapse formation between infected cells. We derived the stationary points of these

models, and evaluated local stability under a variety of parametric conditions.

When realistic parameter values identified from clinical data were used, our

model showed that steady-state viral load of a synapse forming virus increased mono-

tonically with the probability of successful infection r. Steady-state viral load also

increased with the number of viruses transmitted per synapse s until this reached a

threshold measured as a fraction of total burst size B of approximated 1-2% for the

parameters used in this study. This is consistent with the results reported in [170],

and reflects the fact that once the increased probability of successful infection begins

72

to saturate, additional viruses transmitted via the cell-cell pathway are ’wasted”, in

that they reduce the number of viruses transmitted by the cell-free pathway without

significantly increasing the probability of success of the cell-cell pathway. However, by

using realistic paramter values, our model makes it clear that 1-2% of the burst size is

approximately 20-40 viruses, which is on the same order of magnitude as the observed

transmission numbers for synapses in vitro [167]. Figure 7.3 illustrates the fact that a

much higher penalty is paid (measured in viral load at equilibrium) for values of sB

that

are smaller than optimal rather than for values larger than optimal. It is feasible that

the synaptic muliplicity of infection s has evolved to a value larger than the optimum

due to decreased sensitivity of the viral fitness around this level.

When the synapse-forming HIV variant was considered in competition with a

non synapse-forming variant, we were able to derive conditions for the invasion of a

synapse-forming HIV variant against an established non synapse-forming variant. This

condition was best expressed as an upper-bound on the number of viruses transmitted

via a synapse as a fraction of total burst size sB

. The upper bound was most sensitive

to the fitness of the cell-free virus R0f , and was as high as 90% for R0f ∼ 2, which is

consistent with values measured during chronic infection [172, 176], to as low as 50%

for R0f ∼ 10, which is consistent with values measured during acute infection [177]. It

is important to remember that these are not the optimal muliplicities of infection; as

discussed previously, the optimal multiplicity of infection as a fraction of burst size is

on the order of 1%.

As mentioned in the introduction, synapse forming is a known transmission

strategy of HIV. We have developed a model that shows an evolutionary advantage

for synapse forming virus, and which predicts multiplicities of infection for the cell-

cell transmission pathway that are consistent with those observed in experiment. This

model will serve as the basis for future work investigating the impact of cell-cell trans-

mission on viral persistence during suppressive therapy.

73

Chapter 8

OPTIMAL MULTI-DRUG APPROACHES FOR REDUCTION OF THELATENT POOL IN HIV

Human Immunodeficient Virus (HIV) infection is a wide-spread chronic illness,

affecting over 34 million people, with as many as 2.5 million new infections each year

( [153]). Untreated infection results in the progressive depletion of the helper T-cell

population, and the resulting immunodeficiency leads to death by opportunistic infec-

tion ( [152]).

The advent of multi-drug approaches to treating HIV infection, known as combi-

nation antiretroviral therapy (cART), has resulted in HIV infection becoming a chronic,

manageable disease. The durability of viral suppression in some patients has led to

hopes that eradication of the virus and a “cure” for the disease may be possible ( [178]).

When cART is interrupted, rapid viral rebound occurs in almost all patients,

regardless of the duration of virus suppression ( [179]). This is most commonly at-

tributed to the ability of HIV to infect cells without entering active replication. If the

infected cell then takes on a memory phenotype, the latently infected cell can persist

for decades without triggering an immune response, but may re-activate at any time,

triggering a rebound of active infection ( [180–182]).

The decision point where an infected cell either becomes actively infected or

remains quiescent is stochastically determined by a feed-forward process in the tran-

scription of HIV RNA from integrated viral DNA ( [183,184]). When RNA polymerase

binds to the HIV promoter region, transcription usually fails to complete unless the

HIV viral protein Tat binds to the emerging complex ( [185–187]). Once a single RNA

transcript is completed, large numbers of Tat are produced and this binding becomes

74

certain. Conceivably, a drug that acted in a similar manner to Tat could bias this

decision and prevent the formation of latently infected cells, as illustrated in Figure 1.

Lysis

Latent

New Infection

Cell’s DNA

Cell+Viral DNA

Free Virus

Release of Viral RNA

TAT

Extracellular TAT

Lysis New Infection

Figure 8.1: The effect of additional Tat on viral latency. Following cell entry,reverse transcription, and integration, an infected cell becomes either la-tent or actively infected depending on whether intracellular Tat binds theearly transcription product. Exogenous sources of Tat bias this towardactive infection.

Current approaches to the eradication of the viral reservoir formed by these la-

tent cells have focused on a so-called “shock-and-kill” strategy, where immune-stimulating

agents trigger latently infected cells to begin active production of the virus. This al-

lows the cells to become targets of the immune system for killing, or to be killed by the

cytopathic effects of the budding virus ( [178, 188–193]). Commonly discussed agents

include interferon-type drugs, which are generalized immune activators ( [194]).

All of the “shock-and-kill” strategies are predicated on the assumption that suc-

cessful virus replication is essentially halted by the background regimen of cART, and

that the formation of new quiescently infected cells during the viral burst following

activation is essentially impossible. Recent work,however, has shown that anatomical

reservoirs with limited antiviral activity may serve as sanctuary sites, and permit signif-

icant amounts of efficient viral replication during apparently effective cART treatment

( [195–198]). This so-called Cryptic Viremia may create a condition where reservoir

flushing can actually increase the size of the latent reservoir.

In Section 2 of this chapter, we introduce a new model of HIV infection dynamics

that incorporates the activity of two potential drugs targeting the latent reservoir.

75

The first drug is an interferon-like drug which acts by increasing the activation rate of

latently infected cells. The second drug is a Tat analog, which acts by decreasing the

likelihood of infection events leading to the formation of latently infected cells. We find

the bifurcation points for this model where interferon therapy along is not able to clear

the latent reservoir. In Section 3, we simulate the behavoir of the model using model

parameter values derived from the existing literature, and use a simple optimal control

formulation to explore the usefulness of a hypothetical second drug, the Tat analog, in

the presence and absence of cryptic viremia. We show that, while drugs that increase

the activation rate are sufficient to clear the reservoir when cryptic viremia is absent,

drugs that inhibit the establishment of latent infection must be used together with the

activation rate enhancing drugs to achieve clearance when cryptic viremia is present.

8.1 HIV Model

Our HIV dynamic model is based on the extensively studied model of HIV

infection first introduced by [173]. In this model the behavior of uninfected cells,

infected cells and HIV virus is modeled by the equations:

T = λ︸︷︷︸T-cell

Production

− dT T︸︷︷︸T-cell

Death

− β T V︸ ︷︷ ︸Free Virus

Infection

(8.1a)

I = β T V︸ ︷︷ ︸Free Virus

Infection

− dI I︸︷︷︸Infected Cell

Death

(8.1b)

V = k I︸︷︷︸Free Virus

Production

− dV V︸ ︷︷ ︸Free Virus

Death

. (8.1c)

Here T (t), I (t) and V (t) represent uninfected cells, infected cells and virus population

at time t, respectively. The rate of production of uninfected cells is represented by λ.

The death rates of uninfected cells, infected cells and free virus are dT , dI and dV .

The rate of virus production by infected cells is given by k, which can be reduced

by the activity of protease inhibitors. The infection rate is given by β, which can be

76

reduced by the activity of reverse-transcriptase inhibitors and integrase inhibitors. For

the purpose of simplicity, we will model the activity of the background cART regimen

as a reduction in β.

To include latent reservoir, we add latent cell dynamics, as in [199].

6992

6994

6996

6998

7000

T−

cells

0.001 0.01 0.1 1 10 100 1,000

0

20

40

60

Time (days)

Virus load

R0 = 0.999

R0 = 0.5

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Active infe

ction

0.001 0.01 0.1 1 10 100 1,000 10,0000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (days)

Late

nt in

fection

(b)

Figure 8.2: Reservoir Behavior without Treatment. Cryptic viremia (R0 =0.999) provides a mechanism for the maintenance of a steady-state latentreservoir. Without cryptic viremia (R0 = 0.5), the reservoir decays with-out intervention. Results are shown for uninfected T-Cells (a), ActivelyInfected T-Cells (b), Free virus (c), and latently infected T-Cells (d).

T = λ︸︷︷︸T-cell

Production

− dT T︸︷︷︸T-cell

Death

− β T V︸ ︷︷ ︸Free Virus

Infection

(8.2a)

I = (1− ρ) β T V︸ ︷︷ ︸Active

Infection

− dI I︸︷︷︸Infected

Cell Death

+ αL︸︷︷︸Reactivated

Cells

(8.2b)

L = ρβ T V︸ ︷︷ ︸Latent

Infection

− 1

2αL︸︷︷︸

Net

Clearance

(8.2c)

V = k I︸︷︷︸Free Virus

Production

− dV V︸ ︷︷ ︸Free Virus

Death

, (8.2d)

77

with L (t) as the latent cell population, ρ as probability that a new infected cell becomes

latent, and α as the rate of reactivation of an latent cell. Table 8.1 shows all parameters

used in the model and experimentally-derived values for them obtained from [172].

Parameter Value Units Biological meaning

λ 7× 102 cells

µL× dayUninfected birth rate

dT 0.1 1

dayUninfected death rate

dI 1 1

dayInfected death rate

dV 23 1

dayVirus decay rate

k 2× 103 copies

cell× dayVirus copies per cell

β 2× 10−6 mL

copies× dayInfection Rate

α 0.001 1

dayReactivation rate

ρ 0.001 - Latency Probability

Table 8.1: All parameters values were taken from [172].

In this new model, we assume that the net clearance of the latent pool is half

of the reactivation rate of a latent cell. The actual ratio is unknown, as the term 12αL

in Equation 6.2c is the net effect of reservoir cell activation, reservoir cell division,

and reservoir cell death, while αL in Equation 6.2b represents activation only. The

assumption that the net loss of reservoir cells to an activation process is less than the

net production of actively infected cells is consistent with the experimental observation

that the latently infected reservoir is remarkably stable despite relatively high levels of

ongoing activation ( [200]).

The steady state value for the latent pool is

L =2ρ ((1 + ρ) kβλ− dIdTdV )

(1 + ρ)αkβ(8.3)

Note that the latent pool decays to zero if and only if

R0 <1

(1 + ρ), (8.4)

where

R0 =kβλ

dIdTdV(8.5)

78

h=0

h=0.001

h=0.009

0.1 1 10 100 1000Time:Days0

10

20

30

40

50

60

Virus LoadR0=0.5

Time (days)

Viru

s lo

ad

v  

(a)

h=0

h=0.001

h=0.009

0.01 1 100 104Time:Days

0.2

0.4

0.6

0.8

1.0

1.2

1.4Latent infection

R0=0.5

Time (days)

Late

nt in

fect

ion

v  

(b)

h=0

h=0.001

h=0.009

1 10 100 1000 104 105Time:Days

100

150

Virus loadR0=0.999

Time (days)

Viru

s lo

ad

v  

(c)

h=0

h=0.001

h=0.009

1 10 100 1000 104 105Time:Days0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4Latent infection

R0=0.999

Time (days) La

tent

infe

ctio

n

v  

(d)

Figure 8.3: Effect of Interferon Therapy Alone Interferon therapy acceleratesthe decay of the reservoir in the absence of cryptic viremia (a) and (b),but does not change the outcome. In the presence of cryptic viremia (c)and (d), interferon therapy reduces the the steady-state reservoir level,but does not eradicate it. The application of interferon also creates asignificant transient burst of viremia in the presence of cryptic viremia.

is the infectivity ratio of the virus during cART therapy. The baseline probability of

a cell becoming latently infected ρ has been experimentally estimated at about 0.001

( [200]), so a stable steady-state value of the latent pool can only occur when R0 is

very close to 1. We have shown, however, that sanctuary site dynamics can enforce

exactly this condition in the presence of cryptic viremia ( [195]), the presence of which

has been experimentally verified ( [201]). In this chapter, therefore, we will consider

both the condition where cryptic viremia is not present (Inequality 4 holds), or where

cryptic viremia is present (Inequality 6.4 is violated). In all cases, we will assume that

suppressive therapy is present, and that R0 < 1.

79

We also want to model the activity of our two reservoir-targeting drugs. The

interferon-like drugs enhance the activation rate of quiescently infected cells. While it

is possible that the drugs will asymmetrically affect the various factors that play into

the meaning of α in Equations 6.2b and 6.2c, the simplest assumption is that the drug

affects them equally, and this effect can be modeled by replacing α in Equations 6.2b

and 6.2c with (α + η(t)), where η(t) is the administration of the interferon-like drug.

Modeling the effect of a Tat-analog drug on the probability of latency is slightly

more complex. If we assume that the likelihood of any given Tat molecule present

in the cell is an independent binary random process, and that one Tat molecule is

transmitted with each successful infection event, then the effect of adding a Tat analog

at a concentration that increases the average number of Tat molecules per cell by λ(t)

can be written as:

ρ(t) = 1−∞∑n=0

(1− ρ1+n

)Pλ(t) (n) (8.6)

where Pλ(n) is a Poisson distribution with mean λ (t). As a practical matter,

λ(t) = 1 is sufficient to render new latent infections vanishingly improbable.

The dynamics of this model in the absence of any reservoir targeted therapy

(λ(t) = η(t) = 0) is shown in Figure 6.2, using the parameter values shown in Table 1,

with β chosen such that R0 is either equal to 0.5 (no cryptic viremia) or 0.999 (cryptic

viremia). It is clear that when R0 = 0.999, the latent reservoir reaches a steady-state

of approximately 1.4 latently infected cells per µL of whole blood, consistent with

levels observed in patients. This also enables the maintenance of a steady-state viral

load, which is also observed in patient on long-term cART therapy. Conversely, when

R0 = 0.5, the latent reservoir continues to decay slowly toward zero, with a half-life of

approximately 5 years.

When we consider the effect of Interferon-like therapy alone, the behavior of the

model is shown in Figure 6.3. When R0 = 0.5, the administration of interferon has

no measurable effect on the virus load, which decays uniformly in all conditions. The

80

addition of the drug does have a measurable effect on the decay rate of the latently

infected population, though not on the outcome; the latent pool decays toward zero

regardless of the value of η(t).

The behavior is significantly different when R0 = 0.999. Under these conditions,

the administration of the interferon-like drug results in a measurable transient increase

in the viral load. The latent cell population decreases following that addition of η,

but reaches a new steady-state level rather than decaying toward zero. Even when

η = 0.009 (equivalent to a 10-times increase over the baseline activation rate), the new

steady-state latent cell population is still approximately 20% of the baseline latent cell

population. This implies that interferon-type therapies along may be unsuccessful in

clearing the latent reservoir when cryptic viremia is present.

R0=0.999,tat=0

R0=0.999,tat=1

R0=0.5,tat=1

1000 2000 5000 1¥ 104 2¥ 104 5¥ 104Time:Days10-8

10-6

10-4

0.01

1

Latent infectionh=0.001

v  

Time (days)

Late

nt in

fect

ion

(a)

v  

Time (days)

Late

nt in

fect

ion R0=0.999,tat=0

R0=0.999,tat=1

R0=0.5,tat=1

1000 10 00050002000 30001500 7000Time:Days10-8

10-6

10-4

0.01

1

Latent infectionh=0.009

v  

(b)

Figure 8.4: Effect of Interferon Therapy with additional Tat The addition of abackground Tat analog drug dramatically changes the outcome when in-terferon is applied. In the presence of cryptic viremia, interferon therapyis now able to eradicate the viral reservoir. Application of a relativelylow dose of interferon (a) results in a slower decline in the reservoir whencompared to a higher dose of interferon (b).

In order to address this limitation of interferon-like therapies, we consider the

addition of a Tat-like drug together with the interferon-like drug. As we mentioned

previously, the effect of additional Tat on the likelihood of latency formation is so

dramatic that an administration of λ(t) = 1 is sufficient to reduce the likelihood of

latency by several orders of magnitude. Simulations of the model behavior with λ(t) =

81

1 are shown in Figure 4. The addition of Tat eliminates the steady-state behavior of

the latent reservoir, which now converges exponentially to zero, at a rate that depends

weakly on the level of applied interferon-like drug.

8.2 Optimal Control and Simulations

In order to further explore the synergism between interferon-like and tat-like

drugs in reservoir clearance, we explored a simple constrained optimal control formula-

tion of reservoir clearance in the presence of cryptic viremia (R0 = 0.999). While it is

true that only the administration of Tat allows for the exponential decay of the latent

reservoir in this case, stochastic extinction may eliminate the reservoir if it is reduced

to a sufficiently low concentration for an extended period of time.

To simulate the goal of stochastic extinction, we formulated an optimization

problem where latent cell trajectories were constrained to drop below 1% of their

initial concentration by the end of the first year of therapy, and to remain below the

1% level between year 1 and year 2 of therapy. We consider fixed-dose schedules of Tat

and interferon, and minimize the application of interferon.

Figure 5(a) shows the results of this optimization for three conditions. When

Tat is applied alone (η(t) = 0), the problem was infeasible no matter how large λ(t)

was allowed to grow. While the administration of Tat causes the latent reservoir to

decay in this case, the half-life is measured in decades. When interferon is applied alone

(λ(t) = 0), the constraint is achievable with a minimum administration of η(t) = 0.238,

or an effect 240 times the baseline. If Tat is administered λ(t) = 1, the constraint is

achievable with an interferon dose of only η(t) = 0.025. Furthermore, the achieved

trajectory continues to decay exponentially toward zero throughout year 2.

The administration of interferon causes a migration of latently infected cells into

the active infection compartment, and the administration of Tat prevents the migration

of actively infected cells back into the latent pool. This asymmetry of action enables

dynamic treatments, with short-term applications of interferon followed by long-term

administration of Tat. This may be desirable due to the poorly tolerated side-effects of

82

interferon therapy. To explore this possibility, we modified the optimal control problem

to minimize a weighted average of the pulse height and pulse width of a single applied

interferon pulse during a constant application of Tat. The trajectories for two different

weighting values are shown in Figure 5(b). Both a short, high intensity application

and a longer, lower intensity application of interferon are able to meet the desired

treatment constraints. This implies that dynamic therapeutic schedules may be of

value, and should be explored further in future work.

Recent findings suggest that cryptic viremia in sanctuary sites may be common

in treated HIV patients. These findings have significant implications for reservoir-

flushing approaches to “cure” HIV infection. In this chapter, we have demonstrated

that reservoir-flushing approaches using interferon-like treatments may be incapable

of clearing the reservoir in the presence of cryptic viremia. We have shown that, in

the presence of cryptic viremia, additional drugs that reduce the probability of latent

infection may be necessary to successfully clear the pool of latently infected cells. We

further demonstrated through simulation that a significant synergism between these

two drugs exists when applied in the presence of cryptic viremia.

The results in this chapter are preliminary, and many known effects have been

neglected for the sake of simplicity. It would be premature to engage in a more complete

exploration of the optimal control problem, as no Tat analog drug with the properties

described in this chapter yet exists. This is primarily because the phenomenon of cryp-

tic viremia has only recently been recognized, and in the absence of cryptic viremia,

such a drug would have no measurable effect. Indeed, the fastest decay of the latent

reservoir was always observed in the case where cryptic viremia was absent, indicat-

ing that cryptic viremia, rather than latency formation, may be the better target for

therapy.

We have modeled the HIV infection dynamics as a single, well-mixed compart-

ment with a uniform R0 throughout. In fact, cryptic viremia is inherently a spatially

heterogeneous phenomenon, with local regions where R0 > 1, and larger regions where

R0 < 1, resulting in average behavior with R0 appearing to be very slightly less than

83

0 100 200 300 400 500 600 700 800 900 1000

10−6

10−4

10−2

100

Late

nt In

fect

ed C

ells

(per

µL)

0 100 200 300 400 500 600 700 800 900 1000

50100150200

Vira

l Loa

d ( p

er m

L)

0 100 200 300 400 500 600 700 800 900 10000

100200300

Time (Days)

Inte

rfero

n(ti

mes

ba

selin

e)

Tat and InterferonTat onlyInterferon OnlyTarget Concentration (1%)

(a)

0 100 200 300 400 500 600 700 800 900 1000

10−3

10−2

10−1

100

Late

nt In

fect

ed C

ells

(per

µL)

0 100 200 300 400 500 600 700 800 900 100050

100150200

Vira

l Loa

d ( p

er m

L)

0 100 200 300 400 500 600 700 800 900 100020406080

100

Time (Days)

Inte

rfero

n(ti

mes

ba

selin

e)

Low Interferon CostHigh Interferon CostTarget Concentration (1%)

(b)

Figure 8.5: Optimal Reduction of Latent Reservoir When constant-dosageschedules are considered (a), the constrained optimization problem isinfeasible when only Tat analog is administered. Application of Tat to-gether with interferon allows the constraint to be achieved with less than1/10 the dose of interferon when compared to interferon alone. Whenpulsed administration of interferon is considered (b), Tat administrationenables short-dose schedules of the interferon to achieve the constraint.

84

one ( [196]). In future work, we will explore a spatially compartmentalized version

of the model to determine how such spatial heterogeneity would change the effect of

cryptic viremia on the reservoir flushing approaches.

85

Chapter 9

DISCUSSION

Despite considerable leapfrogging due to single-cell technologies shown in chap-

ter 1 and the insights provided by our frameworks in chapters 2 and 3, we are far

from a full grasp over size regulation and expression homeostasis. For example, coor-

dination of critical cell-cycle events with division cycle in various organisms, and their

mechanistic underpinnings still remain unresolved. Identifying how cell-size control is

applied on genome replication in prokaryotes is also closely linked to gene product con-

centration homeostasis since replication time of a gene of interest affects concentration

profiles of its products over the cell-cycle. Given the physiological relevance of concen-

tration homeostasis, could it be that size control over replication timing is exerted to

ensure a constant concentration? Speculating even further, is it possible that the size

control strategies are a mere consequence of maintaining gene product concentrations?

A theory driven experimental investigation in these issues would certainly be useful.

Further, in context of size homeostasis, how mammalian cells achieve size homeostasis

is beginning to unfold only now. Mechanistic insights into how growth rate is regulated

in these cells are critically required. Since size control is intimately tied with activation

of competence in bacteria upon usage of antibiotics [56], a better understanding of size

homeostasis would go a long way in therapeutics and drive new targets for drugs devel-

opment for tuberculosis [12,57]. Regarding concentration homeostasis, the relationship

between mRNA concentration homeostasis and protein concentration homeostasis is

not established as such; specifically it is not known whether both are simultaneously

exhibited. It can be argued that protein concentration homeostasis can be achieved

regardless of mRNA concentration homeostasis by changing translation and/or pro-

tein degradation rates with cell size. Simultaneous measurements of both mRNA and

86

protein numbers of a given gene of interest would shed light into this issue. Another

question of interest is to ask why some organisms (particularly prokaryotes) do not

seem to have dosage compensation while others do. Additional insights are expected

to be developed as these issues are investigated in organisms other than the current

model ones [58].

In chapter 5, we argue that bacterial debris is detrimental for strains with aggres-

sive infectivity. Our hypothesis proposes that this detriment is due to debris produced

by lysed (dead) cell remains. Is cell debris actually present in nature? In practice,

bacterial debris is hard to detect due to its size, which ranges between phage (nm)

and bacterial size (µ m). Recent experiments using biofilm-forming bacteria suggest

that uninfected cells in the center of the biofilm are surrounded by (possible) dead

cells (debris) on the biofilm surface after phage exposition. This dead cells (debris)

may cause a sink for phages at the surface that further protects uninfected cells at the

biofilm center. However, determining if dead cell debris is actually a sink for phages

remains elusive

An alternative to phage waste due to dead cell debris is the possibility of re-

sistance emergence. This hypothesis states that biofilm communities may be able to

produce bacteria strains that are invisible to the phage specie that is attacking the

community. This alternative explanation is rule out since current experiments show

that, although mutation exists, a significant number of bacteria wild types still survive

to the phage’s attack. Phage waste may be produce due to the multiplicity of infection

being larger in the external layers of the biofilm community where the cell population

is mostly infected cells. Whether this is a significant sink for phages remains unknown.

Although we modeled bacterial debris produced by individual lysed bacteria,

our model can be easily adapted to biofilm communities. For instance, we may assume

that instead of modeling a community of bacterial cells, we have a set of bacterial

communities being attack by a phage strain. Therefore, the effective burst size of a

given bacterial community is given by how aggressive is the virus infection. As observed

in the experiments by [], larger adsorption rates are detrimental for viral production

87

since most of the new progeny is attached to cell debris (or already infected cells). The

rate of scape of the virus from a given community allows the virus to visit other non-

infected communities, increasing the chances of invade the environment. Our model

suggest an optimal foraging strategy in which the virus select the proper adsorption rate

which produce maximum host-exploitation. Additionally, we predict similar optimal

foraging strategies in bacterial community environments less elaborated like clumped

bacteria.

The results obtained from the free and synaptic pathway model developed in

chapter 7 suggest several avenues of future research. We have treated multiplicity of

infection as multiple independent trials in this work; future work will consider more

general formulations for the function f(s), including synergistic and antagonistic be-

havior between the multiple virions invading the cell. We have considered all cells to

have the same viral production rate and the same death rate regardless of multiplicity

of infection. The authors of [170] suggest that virus production rate may scale with

multiplicity of infection; future work will include this possibility as well. Viral fit-

ness is known to decrease dramatically from acute to chronic phase infection, and our

model shows that this significantly changes the optimal multiplicity of infection. Fur-

thermore, the decrease in viral fitness is almost certainly due to an increased immune

response [177], and several authors have suggested that cell-cell transmission evades

certain immune system mechanisms [163]. The implications of this for the evolution

of synapse formation rates over the course of infection will be investigated in future

versions of the model. This last consideration may be related to the emergence of

syncytium-inducing variants of the virus (where membrane fusion becomes so extreme

as to result in the formation of huge non-functional multinucleate cells), which has

been associated with disease progression in the final stages of immune collapse into

AIDS [202].

Our synaptic pathway model for HIV infection captures the impact of this trans-

mission mode to the infected cell population which are actively productive. However,

this virus is capable of create quiescent or latent cells which are not affected by any

88

antiviral treatment. These latent cells are persist for decades before stochastically

switching to a fully activated virus producing infected cell. This situation constitute

a major barrier in eradicating the virus from patients. How synaptic transmission

mode may affect the latent population is still a unknown. Research in this direction

had been proposed in a recent paper submitted. We found that synaptic transmission

may provide an evolutionary advantage by means of modulation of how often the cells

become latent.

89

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