Data from the Experiments Yang Cao Department of ...cs6404/Randomness.pdfThe promoter controls the transcript initiation rate. Each transcript leads to a pulse of protein production

Computational Science and Engineering

Randomness in the CellData from the Experiments

Yang Cao

Department of Computer Science

http://courses.cs.vt.edu/~cs6404


Summary

It has been known for many years that cells show variation even with the same gene and under the same environment. This talk will summarize some important cases (not all) in the lab work on this issue.

1. Nongenetic Variation in Chemotaxis

2. Stochastic Gene Expression in a Single Cell


Chamotaxis


What is Chemotaxis


Chemotaxis Model (see Tongli’s Talk)

Korobkova E, Emonet T, Vilar JMG, et al.From molecular noise to behavioural variability in a single bacterium

NATURE 428, 2004

MORTONFIRTH CJ et al.A free-energy-based stochastic simulation of the Tar receptor complex

JOURNAL OF MOLECULAR BIOLOGY 286 : 1059 1999


Back to Years: 1975~1976

Today’s Topic (First Session):

• Quantitation of the Sensory Response in Bacterial Chemotaxis, J. L. Spudich and D. E. Koshland, PNAS, Vol 72, No. 2 (Feb. 1975), 710-713

• Transient Response to Chemotaxis Stimuli in Escherichia coli, H. C. Berg and P. M. Tedesco, PNAS Vol. 72, No. 8 (Aug. 1975), 3235-3239

• Non-genetic Individuality: chance in the single cell, J. L. Spudich and D. E. Koshland, Nature, Vol 262, August 1976, 467-471


Spudich 1975: Methods

• Bacterium: Salmonella typhimurium LT2

• Cultures were grown at

• centrifuged at for 15 min, kept in ice.

• diluted into cold medium to per ml, incubated for 5 min in a water bath and placed on the gyratory shaker.

• A 0.9-ml aliquot of the bacterial suspension was mixed in less than onesec with 0.1-ml of attractant

• of this mixture was delivered under a coverlip to give a chamberof depth .

• A constantly smooth swimming mutant (ST20) was used to control forcontinued motility in the chamber as well as to determine the dilution factor in the final step.

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A bacteria, which causes typhoid fever that affects 16 million pA bacteria, which causes typhoid fever that affects 16 million people annually eople annually

and causes 600,000 fatalities. It has evolved the ability to spand causes 600,000 fatalities. It has evolved the ability to spread from the read from the

intestine to the deeper tissues of humans, including the liver, intestine to the deeper tissues of humans, including the liver, spleen, and bone spleen, and bone

marrowmarrow

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C30o7103×

lµ8

mµ60


Results


Recovery Curve

Poisson Process?


Background Introduction: Poisson Process

Recall that we have introduced in the review of basic probability theory. Poisson process is a very fundamental random process.

λτ

λτ

λ

λτττ-

-

e p(t) :or

dτe ))d ,[ P(t

=

=+∈

dt dt)) t[t, in occurs P(R λ=+

Assumption Probability for one event to occur in a short interval is (approx) proportional to interval length.

The time for next event to occur follows the exponential distribution

Then the total number of such events during time [0 t) follows a Poisson distribution

( ) ( )n!

t t

nN Pλλ −

== en

Can be used to model: • Start time of Phone calls in Palo Alto• Session initiation times (ftp/web servers)• Number of radioactive emissions (or photons) • Fusing of light bulbs, number of accidents• Historically, used to model packets (massages) arriving at a network switch


Additivity of Recovery Time

What does this imply?


Data Fitting for Recovery Time


E. Coli Chemotaxis

• A cell moves along a relatively straight path (runs)

• It stops and jiggles about (twiddles) and then runs

• Twiddles occur at random, about once a second

• The chemotaxis behavior of a cell can be characterized in terms of a single

parameter: the probability per unit time that a twiddle will occur.

• When a cell swims up a spatial gradient of an attractant, the probability is smaller; when it swims down the gradient, the probability is about the same as it is in an isotropic solution.

• It shows: As the concentration of an attractant increases, the cell

adapts slowly at a constant rate. As the concentration decreases, it adapts rapidly.

• Source: Transient Response to Chemotaxis Stimuli in Escherichia coli, H. C. Berg and

P. M. Tedesco, PNAS Vol. 72, No. 8 (Aug. 1975), 3235-3239


Method

• Wild-type E. Coli strain AW405

• Tethered to glass, can stand 1 hour at room temperature

• Rotation was followed in the tracking microscope in a chamber

• Medium was changed by inserting a tube connected to the inlet pipe into a vial containing the new solution and opening the valve on the vacuum line

for 7 seconds.

• Chamber was 0.4 mm deep, 6mm wide and 18 mm long

• Solutions were drawn through the chamber at a rate of about 0.1 ml/sec via inlet and outlet pipes

• A new solution reaches the center of the chamber in 0.7 seconds, displace

more than 96% of the old solution within 1.7 seconds, more than 99% within 2.7 seconds.


Results

Note here:

CW == Run ,

CCW == Twiddle.

This is different from most other papers.


Basic Response to abrupt changes


Note: the model mentioned above is given by:

where and are constants

Theory or Quantitative Model?

DKCKCT +=

DK

DKCKCT +=

K


Data Fitting for Transition times


Transition Times


Question and Method

• Source: SPUDICH JL, KOSHLAND DE, NON-GENETIC INDIVIDUALITY -

CHANCE IN SINGLE CELL, NATURE 262 (5568): 467-471 1976


Results (22 Tumbly Mutant Individuals)


Wild-Type Recovery Individuality



Genetic Variation? Not Likely


Due to Phase of Cell Cycle? Not Likely


Individual Property? Likely

The variety in the response sensitivity variety shows independence of cell cycle phase

and is a property for each individuals.


Relationship of unstimulated to stimulated behavior


Implications


An Interesting Discussion Related to Cell Cycle


Time for a break?


Summary

It has been known for many years that cells show variation even with the same gene and under the same environment. This talk will summarize some important cases (not all) in the lab work on this issue.

1. Nongenetic Variation in Chemotaxis

2. Stochastic Gene Expression in a Single Cell


From 1997 to 2002

Today’s Topic (Second Session):

1. McAdams HH, Arkin A, Stochastic mechanisms in gene expression, PNAS 94 (3): 814-819 FEB 4 1997

Times Cited: 283

2. Elowitz MB, Levine AJ, Siggia ED, et al. Stochastic gene expression in a single cell SCIENCE 297 (5584): 1183-1186 AUG 16 2002

Times Cited: 257


Common Gene Network

Fig. 1. (A) A common coupled-reaction architecture for transmission of information or control in one link in a genetically coupled regulatory cascade. The promoter controls the transcript initiation rate. Each transcript leads to a pulse of protein production from downstream genes. Signal concentration at any time is determined by the cumulation over time of protein production and losses. The concentration of the effective form of a signal protein is sensed and responded to at its site(s) of action. The active form of protein signals is commonly a multimer; we assume a dimer here. Duplicate operator sites binding the same protein are also a common motif [true of 43% of 76 repressible promoters known in 1991 (1)]. P, Pi, proteins; PRx, promoter for protein x. (B) A representative autoregulating prokaryotic genetic circuit where the protein product controls its promoter. Autoregulation often serves to stabilize protein concentrations in a range that establishes sustained activation (or repression) of several controlled promoters.


Prokaryotic Transcription Initiation Intervals

oc RPRPPR ⇔⇔+

RNAP initiation:

Constant Reaction Rate Assumption: Shea-Ackers model assumes that there

is rapid equilibrium between free RNAP and that bound to the promoter in closed

form. Under these conditions, we can consider transcript initiation as a single

reaction characterized by a single rate constant, which is unchanging over sufficiently

short time intervals.

Exponential Distribution: At any instant, each promoter will have a near-

constant probability of transcript initiation per unit time and therefore an exponential

distribution of the time intervals between successive transcripts.

Thus, the probability for a transcript initiation reaction in the small time interval t is

(1/Tavg) exp(-t/Tavg) t, where t is time and Tavg is the instantaneous time

parameter of the exponential distribution equal to the average transcript initiation

interval. The variance of the exponential distribution is (1/Tavg)2 and the distribution

is highly skewed about the mean.


Protein Number Control

Reaction model (A) and binding state model (B) characterizing sequential competitions between ribosomes and RNase E at two closely located sites on the transcript (denoted BS for binding sites). Binding of either occludes the binding site of the other. After ribosome binding leading to initiation of translation, the competition recurs after a delay while the translating ribosome's footprint clears the two sites. This process repeats until RNase E binds and initiates degradation of the transcript. Each competition is an independent event with a probabilistic outcome. A transcript is initially in state 1 and thereafter in one of the five states shown in B. The number of proteins produced, N, will be the number of times state 4 is traversed before the process terminates in state 5. When the system is in any state i, aij dtis the probability of transition to state j in time interval {t, t + dt}, where i and j each denotes one of the states {1, ... , 5}. Observations (see text) suggest that a24, a12 a21 and a35, a13 a31. When the system is in state 1, the probability of another protein is approximately (a12 a24)/(a12 + a13)(a21 + a24), neglecting higher order transitions such as 1 2 1 2 4.


Number of Protein Molecules Produced per Transcript

At each exposure of the ribosome binding site and RNase E sites on the

mRNA, there is a direct competition between ribosome and RNase E

binding. This competition leads either to successful translation and

production of a protein or to degradation or inactivation of the transcript.

Assume independent trials with constant probability p of "success“ for ribosome

binding, the distribution for the number of proteins produced will be P(n = N) = pN(1-

p), where n is the number of proteins from a transcript, in each trial. This is the

geometric distribution function with the mean Navg = p/(1-p). Thus, for example, if Navg

is 10 proteins, then p = 0.91; i.e., the ribosome binds in about 91% of the opportunities.

The geometric distribution is also highly skewed; the variance is p/(1-p)2 and P(n

>= N) = pN. For Navg = 10 proteins, 25 or more proteins will be produced from 9% of

the transcripts. Letting TD be the average time interval between successive

competitions, then the number of mRNA messages Nmsg, surviving in the population

versus time after transcription is blocked would be Nmsg = N0msg

·pt/TD. This is equivalent

to exponential message decay with half life Thalf = (ln(2)/ln(p))·TD.


Combine Them?

• The combination of the exponential time distribution of transcripts and the geometrically distributed number of proteins per transcript largely determines the time pattern of protein production initiated at a single promoter

• The simulated switching delay is the time required in each run to accumulate the necessary concentration of proteins in their effective form to activate or repress the controlled promoter. Most switching in bacterial regulatory networks must be accomplished by a few tens of molecules, since more than 80% of E. coli genes express fewer than 100 copies of their protein product per cell cycle.

• Both long intervals with few proteins and bursts of many proteins in a short time are common occurrences. Consequently, the concentration growth profile in each cell can be quite erratic and distinctive. Significantly, single "bursts" of signal proteins can occasionally be large enough to immediately activate or repress the controlled promoters.


Using Gillespie Algorithm

(A) Three simulation runs for the onset of P1 dimer production for the regulatory configuration in Fig. 1B. Each run is a different realization of the pattern of the dimer concentration growth in an individual cell. The pattern of protein expression can be quite erratic and thus dramatically different in each cell. Rapid changes in dimer concentration due to forward and reverse dimer transitions contribute to the high frequency noise in the protein dimer signal. The broken lines are the declining concentrations equivalent to 25 and 50 dimer molecules in the growing cell. Parameters: P1 dimerization equilibrium constant = 20 nM; dimerization kr= 0.5 s 1; P1 half-life, 30 min. Initial cell volume comparable to E. coli of 1 × 10 15 liters, doubling with linear growth (20) in 45 min (12).

(B) Mean and ± 1 results for 100 runs at gene dosages of 1, 2, and 4. The sigma values plotted are the 16th and 84th concentration percentiles at each time point. At higher gene dosages, protein P1 is being produced from more genes; the concentration rises more rapidly, and the effective concentration range will be reached quicker. In addition, the dispersion in time to effectiveness (i.e., the switching delay) will be lower for faster growing signals.

(C) Activation level of a controlled promoter (e.g., PRP3 in Fig. 1)assuming activation, A, is characterized by the Hill equation with Hill coefficient 2: A = (Kh[P1P1]2)/(1 + Kh[P1P1]2) where [P1P1] is the P1 dimer concentration and Kh is the Hill association constant, Kh = [KE] 2. Curves are labeled by N;KE, where N is the gene dosage and KE is the dimer-operator binding constant. Each curve reflects only the mean concentration curve plotted in B. Activation (or repression) of controlled genes in each cell and over the population will differ widely around this mean value as shown in A and B.


Switch Delay and Lambda Phage

Regulation Facts: Twenty-one (68%) of the

principal 31 regulatory proteins repress their

own synthesis. Four (13%) of the 31 activate

their own synthesis. Four of the regulatory

proteins repress their own synthesis, but

activate other genes. One represses its own

synthesis from one promoter, but activates it

from another promoter.


From Other People’s Data


Another Experimental Work

• Source: Elowitz MB, Levine AJ, Siggia ED, et al. Stochastic gene expression

in a single cell SCIENCE 297 (5584): 1183-1186 AUG 16 2002


Intrinsic and extrinsic noise

Intrinsic and extrinsic noise can be measured

and distinguished with two genes (cfp,

shown in green; yfp, shown in red)

controlled by identical regulatory

sequences. Cells with the same amount

of each protein appear yellow, whereas

cells expressing more of one fluorescent

protein than the other appear red or

green. (A) In the absence of intrinsic

noise, the two fluorescent proteins

fluctuate in a correlated fashion over time

in a single cell (left). Thus, in a population,

each cell will have the same amount of

both proteins, although that amount will

differ from cell to cell because of extrinsic

noise (right). (B) Expression of the two

genes may become uncorrelated in

individual cells because of intrinsic noise

(left), giving rise to a population in which

some cells express more of one

fluorescent protein than the other.

Intrinsic and extrinsic noise are defined by


Experimental Results






Model and Analysis


Simulation Results


Conclusion


Future Work:


Thanks! Questions? Plato is my friend, Aristotle

is my friend, but my best

friend is truth --- Newton

Documents

Data from the Experiments Yang Cao Department of ...cs6404/Randomness.pdfThe promoter controls the transcript initiation rate. Each transcript leads to a pulse of protein production