How and Why to Build a Mathematical Model: A Case Study Using … · 2020. 1. 31. · HowandWhytoBuildaMathematicalModel canbeusedtodevelopamathematicalmodel. We then demonstrate

How and Why to Build a Mathematical Model

How and Why to Build a Mathematical Model: A Case Study Using Prion Aggregation

Mikahl Banwarth-Kuhn1, Suzanne Sindi1

From the 1Department of Applied Mathematics, School of Natural Sciences, University of CaliforniaMerced, Merced, CA, USA

Running title: How and Why to Build a Mathematical Model

To whom correspondence should be addressed: Suzanne Sindi, 1-209-228-4224 [email protected]

Keywords: mathematical model, differential equation, Law of Mass Action, protein aggregation

ABSTRACTBiological systems are inherently complex, and

the increasing level of detail with which we are ableto experimentally probe such systems continuallyreveals new complexity. Fortunately, mathematicalmodels are uniquely positioned to provide a toolamenable for rigorous analysis, hypothesis genera-tion, and connecting results from isolated in vitroexperiments with results from in vivo and whole or-ganism studies. However, developing useful math-ematical models is challenging because of the of-ten different domains of knowledge required in bothmath and biology. In this work, we endeavor toprovide a useful guide for researchers interested inincorporating mathematical modeling into their sci-entific process.

We advocate for the use of conceptual diagramsas a starting place to anchor researchers from bothdomains. These diagrams are useful for simplifyingthe biological process in question and distinguishingthe essential components. Not only do they serve asthe basis for developing a variety of mathematicalmodels but they ensure that any mathematical for-mulation of the biological system is led primarilyby scientific questions. We provide a specific exam-ple of this process from our own work in studyingprion aggregation to show the power of mathemat-ical models to synergistically interact with experi-ments and push forward biological understanding.Choosing the most suitable model also depends onmany different factors and we consider how to makethese choices based on different scales of biologicalorganization and available data. We close by dis-cussing the many opportunities that abound for both

experimentalists and modelers to take advantage ofcollaborative work in this field.

INTRODUCTIONHow living organisms function and biolog-

ical systems work remain some of the most pro-found mysteries in our world. Many areas of bio-logical inquiry aim to explain how complex behav-iors emerge from simple rules of interaction betweenfundamental components. Two examples of this typeof question include, “How do collections of differ-ent cell types communicate to develop into complexstructures such as an entire organ or embryo?” or“How do microscopic protein aggregation dynam-ics and macroscopic cell behaviors within a growingtissue interact to propagate or clear neurodegenera-tive diseases such as Alzheimer’s?" Answering thesequestions is challenging, and requires combining ap-proaches from different domains. In particular, oneof the most successful combinations of expertise isthat of experimental and biological sciences with themathematical and computational sciences. The useofmathematical models was proven to be fundamen-tal towards advancing physics in the 20th century,and many are projecting mathematics to play a simi-lar role in advancing biological discovery in the 21stcentury (1–3).

Mathematical and computational modelshave already begun to play an increasingly large rolein the advancement of biological and biochemicalresearch because they make it possible to quanti-

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tatively bridge the gap between data gathering andmechanism testing by providing a set of analyticaland numerical tools. The level at which we are ableto collect data ranges acrossmany different scales in-cluding single-cell analysis, molecular interactions,tumor growth and epidemiological statistics. Inmostcases, we aim to understand how the processes andmechanisms we track at microscopic scales lead toemergent patterns of disease and other behaviors ob-served at the macroscopic scale of entire colonies,tissues or populations. The aim of combined exper-imental investigation and mathematical and compu-tational modeling in the biological sciences is to de-velop a tool set that can combine information at mul-tiple scales and elucidate the underlyingmechanismsthat drive an observed phenomenon (Figure 1).

Experimental studies are rooted in obser-vation to draw conclusions about how a biologicalsystem functions, and they rely on experimental ma-nipulations to clarifywhich components are themostimportant. However, it can be challenging to makethe required observations andmanipulations withoutaffecting the system in unintentional ways. Mathe-matical and computational models use observationand manipulations in the same way as experiments,but they can avoid many of the most challenging ex-perimental difficulties. In addition, such models of-ten permit experiments that are currently not feasiblein the physical system. In the best cases, mathemati-cal models complement experimental studies by pro-viding new insight on the most crucial interactionswithin the system. Although mathematical descrip-tions of biological phenomena cannot replace bio-logical experiments or provide biological evidencethat a particular mechanisms is at work, they candemonstrate whether or not a proposed mechanismis sufficient to produce an observed phenomenon.

Before access to powerful computers wasso common, it was possible to create and analyzea mathematical model by hand. However, using ahand written equation to study realistic biologicalapplications, such as interactions between meaning-ful numbers of cells or many different biochemicalspecies, becomes impractical due to the large num-ber of equations and variables. Once computersbecame available, code could be written to auto-matically construct and solve equations repeatedlyover multiple time steps. Thus, computational im-

plementation of models is an important feature thatbrings together different kinds of data across a rangeof length scales. Moreover, reasonable knowledgeof computational methods provides a new way forresearchers to investigate interactions between dif-ferent system elements on a very large scale. Evenin the case of biological mechanisms that are wellunderstood, mathematical and computational mod-eling makes it possible to explore the consequencesof manipulating various parameters, which in thecase of brain tumors and other cancers has becomea good resource for simulating different treatmentprotocols before applying them in practice (4–11).

The purpose of this article is to empowerresearchers in the biological and experimental sci-ences to develop their own models as well as tolower the barrier between these fields with the math-ematical sciences. Indeed, there are significant chal-lenges in interdisciplinary collaborations. Both par-ties - the mathematical/computational and experi-mental/biological may feel that they do not havesufficient expertise in the opposite domain to be-gin a fruitful conversation. To paraphrase, JamesA. Yorke - an applied mathematician credited withcoining the term “Chaos Theory” andwho has a longtrack record of productive interdisciplinary work -“interdisciplinary research requires being comfort-able asking ‘naive’ questions." The purpose of thisreview is to help communicate some of the languagesurrounding mathematical modeling in a way thatwill facilitate productive interactions between scien-tists, and to demonstrate that for both sides, trekkinginto the great unknown is not only intellectually re-warding, but offers the potential to introduce signif-icant advances in both fields.

We structure this Review around illustrat-ing a pipeline for mathematical model building inan interdisciplinary setting that has been produc-tive for us in the past. First, we give an overviewof mathematical modeling in general and provide athree-step process we have found useful in modeldevelopment. We then illustrate this process using acanonical enzyme substrate reaction as a guide. InSection 3 we discuss a more complex biochemicalsystem, the aggregation and fragmentation dynam-ics of prion aggregates. Although this system con-sists of a (theoretically) infinite number of interact-ing biochemical species, the same three-step process

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can be used to develop a mathematical model. Wethen demonstrate how this model provides a tool-box for answering scientific questions about priondisease. Finally, we close with a discussion of con-siderations for developing more complex models atdifferent scales and provide resources for the readerinterested in building their ownmodel. In particular,all the code for the models we develop in this Re-view is available in the Supplementary Informationas an iPythonNotebook. We encourage readers toexplore these models on their own. (We also notethat there are many books and resources delving intothis material in greater detail than possible in thisReview. We encourage those interested in learningmore about mathematical modeling to explore otherreviews with examples from different applicationswritten to guide researchers in building models inbiology (12–15) or textbooks in mathematical biol-ogy (16–22).)

WHAT DOES IT MEAN TO MODEL SOME-THING?

Those new to mathematical modeling maybe curious what it actually means to construct amodel. We emphasize that in order to most ef-fectively drive scientific discovery, a model cannotexist in a vacuum but must be intimately involved inthe experimental process. In Figure 1, we demon-strate some interactions between an experimentalinvestigation and a mathematical and computationalmodel. We encourage experimental researchers tothink of a mathematical model itself as an experi-mental tool. The mathematical model can be usedto make a prediction about what will happen undercertain conditions (i.e. different initial conditionsand/or parameters) and observe whether or not thepredicted outcome occurs. The mathematical modelcan be used to study how sensitive one output ofinterest is to increasing or decreasing the amount ofother factors. The mathematical model can also beused to aid in the design of new experiments. How-ever, this process is not one-sided. Developing auseful mathematical model relies on having a solidunderstanding of the system under study. Unknownparameters may be “fit” by comparing model outputto data and the model itself can be “validated” bypredicting an unexpected experimental outcome.

We emphasize three steps in the design of

a mathematical model. The first step in buildinga mathematical and computational model is to for-mulate a diagram that specifies the key players (statevariables) and describes all possible ways these vari-ables might interact with each other. For example,one of the most basic enzymatic reactions first pro-posed byMichaelis andMenten (23), involves a sub-strate S reacting with an enzyme E to form an en-zyme substrate complex E : S which is convertedinto a product P and the enzyme. The diagram forthis system is given in Figure 2 (Left). Note thatthis diagram is by nature conceptual, we are typ-ically specifying only the interactions and not theexact quantitative nature of those interactions. Weemphasize that the creation of this type of diagramis particularly important in interdisciplinary workbecause its visual nature makes it accessible to allresearchers and provides a common ground for mov-ing forward.

The second step in building a mathematicaland computationalmodel is to explicitlywrite out thequantitative form of all interactions. This is shownfor our example system in Figure 2 (Middle). In ourexperience studying biological systems at themolec-ular scale, biochemical equations provide a naturalway to list these interactions. (For a discussion ofhow this step is handled at different biological scalessee Section 4.) It is also at this step that decisionsabout stoichiometry and the form of interactions areclarified (it is possible that complex interactions aresimplified and multi-step reactions determined) andrates (more so as symbols than numerical values)are assigned to specific reactions. Most importantly,this step requires gathering knowledge from experi-ments and experts in the field to incorporate what iscurrently understood about the system of interest.

In our example we see that the enzyme (E)and substrate (S) form a complex (E : S) at ratek1 but that this complex itself is reversible at ratek−1. This complex may also, at rate k2, result inthe creation of a product (P) and a return of enzymein the complex to the free pool. (In this example,the system as defined does not include synthesis ordegradation, so mass is preserved. This simplifica-tion is reflected in our biochemical equations sincethere is no synthesis or degradation and we onlyshow three reaction rates.)

The third step in constructing a mathemati-

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cal model is to convert the explicit quantitative inter-actions into a mathematical framework. In generalthere are many questions to ask before completingthis step. Are the biochemical species consideredto be concentrations or numbers? Is the time-scalediscrete or continuous? Is the system determinis-tic or stochastic? The resulting mathematical for-mulation must include a clear explanation of whichsystem components are being modeled, how eachcomponent is represented (i.e integer, real number,Boolean etc.), and why the choice of representationis appropriate for each component. (For simplicityand because of their popularity, in this paper we willconsider a deterministic model of concentrations.We discuss other possibilities in Section 4.)

How can we convert our conceptual modelinto a mathematical framework? For differentialequation models that aim to capture molecular dy-namics, the standard approach is to use The Law ofMass Action (see Figure 3). This law states that therate of a reaction is proportional to the product ofthe concentrations of the reactants. It is this rate,the product of the reaction rate and the product ofthe reactants, which determines the instantaneouschange in concentration of all biochemical species.In general, the mass action assumption provides atool for setting up a general mathematical frameworkby defining the relationship between state variables(i.e. proportional) in terms of measurable quantities(i.e. concentration of the reactants). In our example,the biochemical reactions lead to the well-studiedsystem of differential equations depicted in Figure 2(Right). (We note that while the Law of Mass Ac-tion was developed for molecular processes, it hasbeen used as a model for interactions at larger scalesand indeed is the foundation for many mathematicalmodels in ecology and epidemiology (19–22).)

After the model is built, the real work be-gins! The model may be analyzed theoretically (byconsidering the form of the differential equationsdirectly) or numerically (by considering the time-varying changes in the quantities of interest). Thefirst task is typically to investigate if the model be-haves in a manner consistent with the knowledgeof the system. For example, the enzyme-substratesystem we considered has been studied extensively.Over time, all of the substrate will be converted toproduct and the shape of the product curve can be

used to determine relationships between the param-eters. Once the model is considered consistent withwhat is known about a system, the model is thenprobed to reveal aspects that would be difficult orimpossible to do experimentally. For example, totest the sensitivity of the system to a changing ini-tial conditions a researcher could set up thousandsof experiments each with different conditions. Al-ternatively, the mathematical model could be moreeasily probed for parameter sensitivity. In addition,it is often only possible to experimentally observe asubset of the state variables of interest (i.e., usuallyonly the product concentration P(t) can be visual-ized). However, the mathematical model can revealthe “hidden” concentrations of all other variablestoo. Because we suspect most readers are familiarwith this simple system, in the next section we ex-plore developing a mathematical model in a morecomplicated setting and demonstrate how analyticaland numerical methods are used to study the model.

CASE STUDY: PROTEIN AGGREGATIONKI-NETICS

In the previous section we went over thebasics of building a mathematical model through:(1) Creating a diagram of the critical players (statevariables) and their interactions; (2) Enumeratingthe complete set of interactions, typically as a set ofbiochemical equations; (3) Translating these interac-tions into a mathematical model, typically throughthe Law of Mass Action. Because we suspect manyreaders are familiar with Michaelis-Menten Kinet-ics, we now consider the same modeling guidelinesto construct a mathematical model used in the studyof prion aggregate dynamics.

Prions are associated with a number of pro-gressive, incurable and fatal neurological diseasein mammals (24, 25). Remarkably, these diseasemay be either spontaneous, genetic or acquired (26–28). These different modes of transmission, andextremely long incubation times, made it challeng-ing for researchers to determine the infectious agent(29–31). However, today the scientific consensusis that they are caused by a protineasous infectiousagent, which is where the term prion comes from(32–35). (See (36) for detailed history on the studyof prion disease.) While the true biochemical andbiophysical processes of prion aggregate dynamics

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are highly complicated, andmay differ depending onstrain and infected tissue, mathematical modeling ofidealized biochemical processes has been extremelyeffective at uncovering critical steps in the pathway(37–41). Before describing a mathematical frame-work for modeling the dynamics of prion proteins,we consider what is known about the biological sys-tem. We will then go through the underlying de-velopment of a diagram identifying the key playersand their interactions. Thenwewill discuss amathe-maticalmodel based on the biochemistry and discusswhat it enables us to learn about the biology.

All mammalian prion diseases are a conse-quence of the same protein PrP (prion-domain pro-tein). In these diseases, a misfolded form of thePrP (PrP-C) is introduced to the host and, ratherthe be eliminated or cleared by cellular quality con-trol mechanisms, this misfolded (prion) form per-sists and induces other normally folded proteinsin the host to fold into their same conformation(42). This conversion process is thought to hap-pen through associations between normally foldedprotein and aggregates of the prion form. The sizeof these prion aggregates thus increases by incor-porating the newly misfolded protein (polymeriza-tion). The size of these aggregates may also de-crease either through depolymerization (which re-moves a single monomer) or fragmentation (whichthen can amplify the total number of aggregates).These aggregates then spread through the mam-malian host. Remarkably, prion diseases are notonly isolated to mammals. The single cell fungiSaccharomyces cerevisiae has several harmless phe-notypes which we now know to be caused by prionproteins (34). Moreover, certain human neurode-generative diseases (Parkinsons’s, Huntington’s, etc)propagate by a similar aggregation process and, aswas demonstrated by a recent study in Alzheimer’s(43). Finally, although prion aggregates are ex-tremely thermodynamically stable, the appearanceof prion disease is thought to be nucleation depen-dent (44). That is, the aggregates can be thoughtof as a protein phase transition that necessitates theformation of a stable nucleus consisting of somenumber, n0, of misfolded protein monomers (45).(For the [PSI+] prion in Saccharomyces cerevisiaethe nucleus is thought to consist of 5 misfoldedmonomers (37, 40).) Aggregates below n0 in size are

highly unstable and are thought to rapidly resolvedinto monomers(35, 40, 44, 46)).

Setting the Stage for a MathematicalModel—After having reviewed the biological back-ground on prion aggregation, the next step is to deter-mine the key biochemical players and how they willinteract. This step is particularly important as it setsthe stage for how complex our model will ultimatelybecome. In the case of prion dynamics, we know thatthere are a host of biochemical factors that are impor-tant for disease propagation. But it is usually best tobegin with the simplest set of assumptions and addcomplexity only when needed. (For a discussion ofhow to further develop our mathematical model toinclude more complex interactions see Section 4.).In what follows, we lead the reader through the de-velopment of the Nucleated Polymerization Model(NPM). This model has become the standard choicefor modeling prion aggregation dynamics (44, 46).First, based on the biology we know, we need toconsider two classes of biochemical species: solu-ble protein monomers and aggregates of the prion(misfolded) form of the protein. Because we knowaggregates consist of groups of misfolded proteins,we will consider aggregates of every discrete sizelarger than the critical nucleus size n0.

The next question we ask is how will thesespecies interact? We know that there is some pro-cess by which normal protein (a soluble proteinmonomer) is converted to the prion form and incor-porated into an existing aggregate, and some otherprocess by which aggregates are fragmented. Sowe will need to specify these interactions. In ad-dition, following the common critical nucleus sizeassumption any aggregate below the nucleus sizen0 in our model will be considered not stable andquickly resolved into healthy monomer. As we saidbefore, once the species and interactions are iden-tified, it is advisable to codify these properties ina picture or diagram. In addition to being a usefultool for a mathematical modeler on their own, the vi-sual medium of pictures facilitates communicationbetween researchers in different domains. Agreeingon a diagram of interactions is an important first stepin mathematical modeling.

We show a diagram depicting our prion in-teractions in Figure 4 (Left). We consider two typesof biochemical species - soluble (monomer) and ag-

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gregates. We have also introduced the idea that thenumber of blocks in an aggregate indicates its size(i.e., conversion shows an aggregate of size 3 becom-ing an aggregate of size 4). We have also diagramedthe basic nature of the conversion and fragmentationreactions. Notice that in drawing our fragmenta-tion equation, we had to make a decision about whathappens when fragmentation creates an aggregatewith size below that of the nucleus (n0 = 2). Whileone choice could be that those proteins completelydegrade, we have instead decided to have these pro-teins return to the soluble monomer state. (This isconsistent with the NPM (44, 46).) Our diagramalso depicts two additional reactions: monomer iscreated (rate α) and both monomer and aggregatesare degraded (rate µ). The inclusion of such pro-cesses depends on the system under consideration.(Because the original NPM was developed to depictin vivo prion kinetics we assume that protein synthe-sis and degradation machinery to be functioning. Ofcourse, aggregation models of in vitro aggregationmay elect to consider different processes as in (47).)

Once the diagram is defined, the next stepis to write out the steps in the diagram explicitlyas a series of biochemical equations. (See Figure 4(Middle).) The left hand side of the biochemicalequations give the reactants, what is needed or con-sumed for the reaction to be carried out, and the righthand side shows the resulting products. The reac-tion rate designates the speed at which the reactiontakes place. The units of the reaction rate typicallydepend on the order (number of reactants) of the re-action itself. (For example, in a first order reaction(only a single reactant) the reaction rate has units of(time)−1.)

It is during this step that the details of our in-teractions are refined and formalized. In the case ofour conversion equation, we might consider the sizeof an aggregate to impact the rate of conversion. Forexample, an aggregate of size 10 could more easilyconvert than an aggregate of size 5 if every part ofits surface was capable of templating. In the caseof the NPM, prion aggregates are considered to beamyloid - an ordered linear structure of misfoldedprotein. As such, conversion of monomer is onlyconsidered possible at the ends of the amyloid. Assuch, no matter the size of the aggregate, it has onlytwo sites (either end) where conversion can occur.

This is reflected in the top biochemical equation inFigure 4 (Middle).

A similar decision is necessary for the frag-mentation reaction. It is possible that not all sites ina linear aggregate are equally frangible. (Indeed, insome in vitro experiments the best fitting mathemat-ical model was one in which the middle of amyloidfibers was more likely to fragment than the ends(48).) The assumption taken by the NPM is thateach junction between monomers is equally likely.In this case, if each junction in an aggregate frag-ments at the same rate γ then given an aggregateof size (i + j) there are two fragmentation sites thatwould results in two aggregates with distinct sizes(i and j). (Note that if i = j, then while thereis only one site that could create these aggregates,the stoichiometry remains the same because we pro-duce two aggregates of the same size.) Similar bio-chemical reactions govern the processes of synthesis(α) and degradation (µ). (Note that in the originalNPM, distinct degradation rates were considered formonomers and aggregates. However, for simplicitywe consider both processes to occur ate the samerate µ.)

Development and Analysis of the NucleatedPolymerization Model: A Mathematical Model ofPrion Aggregation—With the first two steps com-plete, the next phase is to design and analyze anappropriate mathematical model. Before decidingon a mathematical framework for our model, weneed to decide the types of scientific questions wewant to answer. In this exercise, we will considertwo questions of interest:

• Can we determine what causes prion aggre-gates to be cleared by manipulating biochem-ical rates?

• Why is there a long incubation time for manyprion diseases?Because these questions are based on the

temporal evolution of prion aggregates, we willchose a differential equation model that tracksthe changes in concentration of each biochemicalspecies (see Figure 4 (Right)). That is, we definex(t) to be the concentration of healthy monomer andyi(t) to be the concentration of an aggregate of size iand track their evolving concentrations in time. Weuse the Law of Mass Action to convert our biochem-ical equations into differential equations. As men-

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tioned above, this law states that the instantaneousrate of a reaction is a product of a reaction rate andthe product of the concentration of all reactants atthat time. Let’s look at an example. Suppose thecurrent concentration of the monomer x and aggre-gates yi are known. What is the instantaneous rate ofaggregate conversion of monomer by an aggregate

of size i?According to the Law of Mass Action and

the top biochemical equation in Figure 4 (Middle),this rate is given by 2β times the concentration ofmonomer x times the concentration aggregates ofsize i, yi:

Rate of Conversion of Monomer by Aggregate of Size i := 2βxyi .(1)

We could repeat the steps for each possiblereaction in our system. (Note because our systemconsiders all aggregate lengths larger than n0, oursystem formally has infinitelymany reactions to con-sider! We will soon see how this is no obstacle formathematical modeling.) We then combine thesereaction rates into a differential equation for eachbiochemical species. The differential equations arethemselves simply the sum of reactions which createthe given species (Rate In)minus the reactionswhichconsume the species (Rate Out). Let’s consider thedifferential equation for the monomer concentration

x(t):dx(t)

dt= Rate In − Rate Out. (2)

Let’s first consider the “Rate Out” part of the equa-tion. There are two types of reactions which con-sume monomer. First, monomer is being degradedat rate µ times the current concentration of x(t).Second, monomer is consumed during conversionby every possible aggregate size. We have alreadycalculated the rate of conversion by an aggregate ofsize i in Equation (1). In calculating this bulk rate weneed to sum the rate of conversion over all possibleaggregate sizes:

Rate Out = µx(t) + 2βx(t)yn0(t) + 2βx(t)yn0+1(t) + . . .

= µx(t) + 2βx(t)∑i≥n0

yi(t).

Now, we move onto the “Rate In” portion ofthe differential equation for x(t). As for the “RateOut” process, there are two types of reactions whichcreate monomer. First, monomer is synthesized atrate α. Second, monomer is created when aggre-gates are fragmented in such a way that one pieceof the recently fragmented aggregate is below theminimum stable size.

To construct the “Rate In" portion of thedifferential equation, we consider an aggregate of

exactly the minimum stable size n0. This aggregatehas n0 − 1 fragmentation sites, and every possiblefragmentation event will result in the creation of twoaggregates with size less than n0. Due to our as-sumption that any aggregate below the nucleus sizen0 is not stable, each of the fragmented pieces willbe resolved into n0 monomers. Thus, the rate ofmonomer creation by fragmentation of aggregatesof size n0 is given by:

Rate of Monomer Creation by Fragmenting an Aggregate of Size n0 := γn0(n0 − 1)yn0(t).

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For an aggregate of size (n0+1) there are n0fragmentation junctions. Two of them will create anaggregate of size n0 and recover a single monomer,and the remaining (n0 − 2) junctions would result inall (n0 + 1) proteins in the aggregate returning to themonomer state. (For example, suppose n0 = 3 andan aggregate of size 4 is going to be fragmented.There are three fragmentation sites, each of whichis equally likely to be chosen. If the first or third

fragmentation site is chosen, the result pieces willbe size 1 and size 3. Since n0 = 2, the size 1 piecewill return to the monomer state and the size 3 piecewill remain an aggregate. If the second fragmenta-tion site is chosen, the resulting pieces will both be2, which is smaller than n0, and the result will be 4monomers.) Thus, the rate of monomer creation byfragmentation is given by:

Rate of Monomer Creation by Fragmenting an Aggregate of Size n0 + 1 := γ (2 + (n0 − 2)(n0 + 1)) yn0+1(t)

= γ(2 + n2

0 − 2n0 + n0 − 2)yn0+1(t) = γ

(n2

0 − n0

)yn0+1(t)

= γn0(n0 − 1)yn0+1(t).

Note that this has the same coefficientγn0(n0 − 1) as before. In fact, this pattern holdsfor all aggregate sizes, but requires slightly differentreasoning when the aggregate size exceeds 2n0. Inthis case, at most one of the pieces resulting froma fragmentation event will be smaller than n0. (Forexample, suppose n0 = 3 and an aggregate of size 6is going to be fragmented. There are 5 sites, eachof which is equally likely to be chosen. If the mid-dle fragmentation site is chosen, the resulting pieceswill each be 3, the same as n0. Every other fragmen-

tation site will result in one piece that is smaller thann0, that will return to the monomer state, and onepiece larger than n0, that will remain an aggregate.The two outermost fragmentation sites will result inrecovering 1 monomer and the last two will recover2 monomers. The sum of monomers recovered, re-gardless of the original aggregate size, will alwaysbe n0(n0 − 1).)

As such, the total “Rate In” portion of thedifferential equation for x(t) is:

Rate In = α + γn0(n0 − 1)yn0(t) + γn0(n0 − 1)yn0+1(t) + . . .

= α + γn0(n0 − 1)∑i≥n0

yi(t).

Combining these together, we have the differential equation for x(t) as follows:

dx(t)dt=

(α + γn0(n0 − 1)

∑i≥n0

yi(t)

)︸︷︷︸

Rate In

−

(µx(t) + 2βx(t)

∑i≥n0

yi(t).

)︸︷︷︸

Rate Out

(3)

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Similar reasoning results in the following differential equations for yn0(t) and yi(t) when i > n0:

dyn0(t)dt

=

(2γ

∑j>n0

yj(t)

)︸︷︷︸

Rate In: Bigger Aggregates Fragment

−

©«2βx(t)yn0(t)︸︷︷︸Conversion

+ γ(n0 − 1)yn0(t)︸︷︷︸Fragmentation

+ µyn0(t)︸︷︷︸Degradation

ª®®®¬︸︷︷︸Rate Out

(4)

dyi(t)dt

=

©«2βx(t)yi−1(t)︸︷︷︸

Conversion

+ 2γ∑j>i

yj(t)︸︷︷︸Fragmentation

ª®®®®®®®¬︸︷︷︸Rate In

−

©«2βx(t)yi(t)︸︷︷︸Conversion

+ γ(i − 1)yi(t)︸︷︷︸Fragmentation

+ µyi(t)︸︷︷︸Degradaton

ª®®®¬︸︷︷︸Rate Out

. (5)

Now that we have out mathematical model,we begin the process of analyzing it. Generally,there are two approaches: analytical and numerical.With analytical approaches, we generally use thestructures of the equations themselves to determinekey characteristics such as steady-states (caseswherethe system configurationwill remain unchanged) andtheir stability (where the systemwill tend over time).With numerical approaches, we use software (Mat-lab, R, Mathematica, Python, etc) to study the tem-poral dynamics the biochemical species in the sys-tem. To use numerical methods we typically needto specify the initial condition of the system and allbiochemical reaction rates (α, β, γ, µ, n0). We willconsider each approach below to answer the twoquestions we posed originally.

Stability of the Prion Aggregate System: An-alytical Approach—Our model (Equations (3)- (5))

may appear quite daunting, but in fact is highlyamenable to analytical methods. We will use thesemethods to consider our first question about whatcauses prion aggregates to be cleared and the dis-ease phenotype to be reversed. We first rearrangethe equations into a form that makes them easier towork with. Rather than considering each aggregatesize yi we will consider two new quantities about theaggregates as follows:

Y (t) =∑i≥n0

yi(t) and Z(t) =∑i≥n0

iyi(t).

Note that Y (t) represents the total concentration ofprion aggregates, while Z(t) represents the concen-tration of total protein in prion aggregates. Remark-ably, our system of differential equations in yi canbe arranged to a set of differential equations forY (t),Z(t) and x(t) (our original monomer population):

dx(t)dt

= α + γn0(n0 − 1)Y (t) − µx(t) − 2βx(t)Y (t) (6)

dY (t)dt

= γZ(t) − (µ + γ(2n0 − 1))Y (t) (7)

dZ(t)dt

= 2βx(t)Y (t) − µZ(t) − γn0(n0 − 1)Y (t). (8)

With only three equations it is much easierto solve for the steady-states of the system. Steady-states represent cases where the system is unchang-ing and so we identify them by finding values of

x,Y, Z that satisfy:

dxdt=

dYdt=

dZdt= 0. (9)

Some algebra will demonstrate that the system has

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exactly two steady-state solution. The first is theaggregate-free steady-state: x = α/µ, Y = Z = 0.In this case, all protein is in the healthy monomerstate. The second nontrivial steady-state (x∗,Y ∗, Z∗)is more cleanly shown in the following relationships:

x∗ =α/µ

R0Z∗ + x∗ = α/µ

Z∗/Y ∗ = 2n0 − 1 +µ

γ

where

R0 =2αβγ

µ(µ + γ(n0 − 1))(µ + γn0). (10)

Those familiar with epidemiology will recognize theuse of the R0 value to represent the basic reproduc-tive number (21). In epidemiology, R0 representsthe number of secondary infections produced by oneprimary infection in a susceptible population. WhenR0 < 1, no matter how many aggregates are initiallypresent, they will all eventually be cleared and thesystem will converge to the aggregate free steady-state. When R0 > 1 the introduction of any initialamount of aggregate will lead the system to convergeto the non-trivial steady-state.

How does this analytical work help us tobetter understand how biochemical rates can be ma-nipulated to clear prion aggregates? Deriving analgebraic expression such as the one given in Equa-tion (10) above is key. We know that when R0changes from a value greater than 1 to a value smallerthan 1, prion aggregates will be completely cleared.But what parameter values should we change in or-der to arrive at such an R0 value? And by howmuch? While it might seem obvious that increas-ing the degradation rate µ above a certain thresholdwill cause prion aggregates to be cleared, lack of analgebraic expression for R0 wouldmake it almost im-possible to determine the precise degree of changein parameters to produce the desired output.

For example, the parameters used to plot thesystem in the first row of Figure 5 below, produce anR0 value of approximately 5.7. As expected, sincethis R0 value is greater than 1, prion aggregates dopersist and reach a positive stable steady-state asshown. However, if we fix all other parameters, andonly change the fragmentation rate γ, we can use

Equation (10) above to show how big or small γmust be to produce an R0 value less than 1 whereprion aggregates are cleared. Keeping all parame-ters the same as Figure 5(A) we have

R0 < 1 =⇒2αβγ

µ(µ + γ(n0 − 1))(µ + γn0)< 1

=⇒ γ > .00330561.

Dynamics of the Prion Aggregate System:Numerical Approach—To answer our second ques-tion, why is there such a long lag time in the man-ifestation of prion disease, we consider numericalsimulations of our prion model. While at one point,the lack of easy to use computational tools made itdifficult for all but a relatively small group of expertsto numerically solve differential equations, this is nolonger the case. There are a multitude of free touse computational tools that enable anyone to nu-merically analyze even sophisticated mathematicalmodels. Of course, as the complexity of the modelgrows, typically more effort is required to numer-ically code the solution. In this work, we use aniPython Notebook to analyze the dynamics of ag-gregates with an eye towards understanding why ittakes so long for prion diseases to manifest.

In Figure 5 we illustrate the concentration ofx(t) and Z(t) (A & B Left) as well as the aggregateconcentration Y (t) (A & B Right) for a particularchoice of biochemical parameters upon the intro-duction of a very small amount of prion aggregateZ(0) = Y (0) = 1 nM. (The biochemical parame-ters were chosen to roughly mirror the propertiesof the [PSI+] weak strain in yeast but plotted ona time-scale relative to mammalian prion disease(40, 49).) In (A) we see that the amount of healthymonomer x(t) decreases relatively slowly; after 400weeks only 9% of the protein has changed to theprion (misfolded) state. This is because, initially,the aggregates are at a very low concentration andthus conversion of monomer is favored over frag-mentation. Because conversion only adds to existingaggregates (but does not create them) this has only amodest impact on the pool of healthy protein. How-ever, just because we can not see much of a decreasein x(t) does not mean we are safe!

Eventually, the aggregates increase enoughin size such that fragmentation becomes significant

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enough to create new aggregates at a discernible rate.This autocatalytic process then increases the rate ofaggregate conversion and eventually produced a no-ticeable decrease in the healthy protein. It is pre-cisely this effect, the initial balance that favors con-version over fragmentation that causes the long-termdelay in the manifestation of prion disease pheno-types. In (B), we doubled the conversion rate, andas a result, the system reaches it’s steady state almosttwice as fast! We next describe how this model wasused by ourselves and previous researchers to learnabout prion aggregation processes.

Interaction Between Mathematical Modelsand Experiments—With a mathematical model inplace, it is now time to begin the process of using itin concert with experiments to study and probe thesystem. One common first step is to fit parametersby relating model output to experimentally observ-able quantities. For example, when the NPM wasintroduced the authors related the biochemical pa-rameters (α, β, γ, µ, n0) to the exponential growthrate in aggregated protein they observed during earlyphases of prion disease through the relationship ofthe parameters to the R0 (44). Similarly, mathemati-cians Greer, Pujo-Menjouet and Webb used priorexperimental studies to fit the kinetic parametersof their model a strain of scrapie (46, 50). Withfit kinetic parameters, models can then make pre-dictions about what should happen under particularinitial conditions, X(0),Y (0), Z(0). Indeed, as pro-tein aggregation modeling has advanced easy-to-usecomputational pipelines have been developed for re-searchers interested in fitting their in vitro aggrega-tion curves (51).

When a model does not match experiments,this typicallymeans themathematical representationis not considering all the relevant behavior of the bi-ological system. Deciding what is missing and howto modify the model is challenging and requires theclose interaction of experimental and mathematicalscientists. However, this is whenmodels can be theirmost useful because they can suggest novel hypothe-ses and experiments. Researchers found that theNPM did not consistently match experimental datafor the [PSI+] prion (49). The only way the authorsfound to simultaneously fit all experimental obser-vations (steady-state levels of soluble protein, aver-age aggregate size and loss of the prion phenotype)

was to add two features to the model: (1) aggregatetransmission during cell division is biased by aggre-gate size and (2) fragmentation depends on a rate-limiting molecular chaperone. This suggested newexperiments that would only be true if the newmodelwas true. First, if aggregate transmission is size de-pendent, the mathematical model predicted that thelevel of soluble protein a cell has should be linkedto its age. This prediction was then experimentallyvalidated, by sorting cells by soluble protein andcounting their bud scars (generational age). Second,if fragmentation is rate-limiting then increasing thesynthesis rate α was predicted by the mathematicalmodel to increase the size of aggregates. (Under theoriginal NPM increasing the synthesis rate α doesnot change the shape of the aggregate distribution.)This prediction was also experimentally validated byusing a different promoter.

DISCUSSIONWe have now seen two examples about

building mathematical models with our three-stepprocess, and highlighted how they can be used tolearn about the molecular processes directing prionaggregation in yeast. In this section, we aim toprovide more insight about how to introduce addedcomplexity into amodeling framework. Specifically,we will discuss several examples of models of pro-tein aggregation dynamics that have been extendedto incorporate processes at distinct biological scales.In addition, the end of this section will discuss someof the challenges in modeling and give additionalcomments on considerations in using mathematicalmodels.

One challenge in mathematical modeling isto determine the right choice of model to answer thescientific questions being posed. Often researchersunderstand that a mathematical or computationalmodel can be a valuable tool, but they may seekto develop a model before establishing exactly whatquestions they want to answer. The choice of modeldepends onmany factors. Starting with several well-defined questions or hypotheses can help determineboth the scales and level of complexity that need tobe considered and lead to an appropriate choice ofmodel. For example, some questions that might beasked include:

• What is the catalytic rate of a particular en-

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zyme reaction?– To address this question, a modelmust minimally include processes at themolecular scale. However, dependingon desired complexity, the choice couldbe made to model the reaction of in-terest as a one step process or includeseveral successive steps in the reaction.In addition, important steps impactingan enzyme reaction could occur at thesame scale or some step might occuron a larger scale such as cell behaviorsor nutrient gradients that impact the en-vironment where the reaction is takingplace.

• Why does my experimental system have suchhigh variance?

– Sensitivity analysis tools offer greatmethods to identify the cause of vari-ance in complex biological systems. Toanswer this question, a model couldconsider one scale only and investi-gate the impact of variability of one ortwo factors at the same scale. Or, themodel could incorporate processes at themolecular scale and processes at the cellor tissue scale and use global sensitivityanalysis methods to identify which fac-tor(s) have themost impact on the overallvariance of the system.

• What are the best time-points to sample myexperimental system to observe dynamics ofthe behavior I wish to study?

– Biological processes in the same systemhappen at different timescales. For ex-ample, chemical reactions happen muchfaster than the time it takes for a cell todivide or move across an agar plate. De-pending on the scientific question beingasked, samples at different time pointscan provide necessary experimental datato calibrate model components at differ-ent scales.

When scientific questions lead model de-velopment, it becomes more clear what processesand scales must be included in the mathematicalmodeling framework and this determines the kindof mathematical model and analysis possible.

Incorporating Processes at Multiple Bio-logical Scales—Biological systems are inherentlymultiscale because all living organisms are com-posed of many different functional networks thatoperate across diverse temporal and spatial scales(Figure 6). Technological advances have made itpossible to attain highly detailed descriptions of keybiological components at almost any scale imagin-able, i.e. genetic mutations, formation of a bloodclot, and even the growing and shortening phasesof microtubules in individual cells (52–57). Sincecutting edge technologymakes it possible to observebiological phenomena at each unique layer of orga-nization, our understanding of how life works is nolonger limited by the instruments in our labs. How-ever, with every new experiment comes additionalcomplexity. The abundance of information availablehas ultimately resulted in the compartmentalizationof scientific inquiry, and led to separate study of eachdifferent biological scale (i.e. molecular biologistsvs. cellular, organismic or population, etc). Mathe-matical and computational models provide a tool tointegrate knowledge from different scales and iden-tify how collective interactions on a fundamentalscale can give rise to a large-scale phenomena (Fig-ure 6). Thus, one challenge in building a mathemat-ical model is determining on what scale (or scales)the key players and interactions occur. One impor-tant question to ask is which scale(s) of resolutionwill provide the most information for understand-ing the underlying mechanisms of the system? Anequally important and converse question is whetherthere is a scale of resolution that offers no insightand should be simplified or ignored?

To further illustrate how to incorporate bi-ological processes at different scales into a model-ing framework, we consider the multiscale natureprion disease dynamics (Figure 6). At the most ba-sic scale, biochemical interactions depicting conver-sion, fragmentation, synthesis and degradation formthe basic mechanisms that lead to the presence ofprion disease (Figure 6 A). For this reason, the ma-jority of mathematical models developed to answerthe question of what causes prion aggregates to becleared have only considered interactions betweencomponents on the molecular scale (see Section 3).However, in some instances, these models were notable to reproduce experimental data. In this case,

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the given modeling framework can be extended toinclude processes at the next level of organization inorder to attempt to produce model results that agreewith experimental data (Figure 6 B).

For yeast prions, the next level of organiza-tion is subcellular. Because molecular chaperoneshave been shown to be essential for prion propaga-tion (49, 58), Davis et al. (59) extended the NPMby modifying the existing biochemical interactionsgiven in Section 3 to account for the impact ofmolec-ular chaperones. The addition of these new bio-chemical interactions resulted in better agreementwith experimental results that could not be sup-ported by the original equations alone. In addition,this work further supported the hypothesis that thereexist many different prion strains. In many cases,the distinction between models that look only at themolecular and/or biochemical scale and models thatincorporate processes on the subcellular scale in-volves adding differential equations to an alreadyexisting modeling framework to describe processessuch as diffusion or spatially varying density func-tions in addition to keeping track of individual par-ticles and/or groups of particles.

The next tier of resolution is the whole cell(Figure 6 C). One of the most unique complicationsintroduced by studying prion disease in yeast, comeswith the consideration that prions in yeast propagatewithin a colony of growing and dividing cells. Thetime it takes for protein to change configurationsis much faster than the time it takes for a cell togrow or divide. Thus, an interesting question toconsider how cellular behaviors such as cell cyclelength, protein segregation at the time of division,and/or age of cells impact protein aggregation dy-namics throughout the entire colony (Figure 6 B andC). To address this question, a new model must in-clude interactions between intracellular componentswithin each individual cell, individual cell behav-iors impacting intracellular dynamics, and cell-cellinteraction between many different cells in the samecolony. One type of well established modelingframework that is capable of integrating molecu-lar, subcellular and cellular level processes is agent-based models. Agent-based models represent cellsas discrete units that interact with each other and canalso carry out individual cell processes such as pro-tein aggregation, division, and growth (for reviews

see (60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65–69)) Fortunately, incorporating processes at differ-ent scales using agent-based models does not re-quire starting from scratch! In the case of yeastprion biology, one of the many well-studied modelsof protein aggregation that focuses on the molecu-lar scale only can be coupled with an agent-basedframework. Thus, developing a model that incor-porated processes at the cellular scale would onlyrequire building an agent-based framework, or rely-ing on one of the many great code bases that exist inthe literature.

At the highest scale, prion disease manifestsas tissue atrophy in mammals and population levelphenotypes in yeast (Figure 6 D). In many studies,population level data is the only quantitative outputavailable from experiments. However, the concen-tration of a protein or other cellular constituent isknown to vary considerably among cells in the samecolony. This heterogeneity is thought to arise fromseveral sources, including differences in kinetic ratesbetween individual cells and distribution of cellularconstituents at the time of cell division. Connectinginteractions between components at the fundamen-tal scale with population level phenotypes is experi-mentally challenging. However, in multiscale mod-els, perturbations of parameters at the fundamentalscale (i.e. protein modifications) can generate ob-servable and measurable changes to coarse-grainedoutputs at the population level ( i.e. colony pheno-type). Integrating processes across different spatialand temporal scales can be achieved using agent-based models described earlier. Furthermore, agent-based models have been used in many applicationsranging from tumor growth, blood clot formation,stem cell regulation in plants and tissue growth andformation to study emergent global patterns basedon local changes to biochemical and molecular pa-rameters or individual cell behaviors. Another classofmodel that can be used to study dynamics of entirecell colonies, tissues and organs are continuousmod-els that use ordinary differential equations or partialdifferential equations to represent the growth of atissue as one continuous sheet, or the change in theshape or position over time of the edge of a colonyor group of cells as one continuous boundary. Inmany cases, discrete, agent-based modeling frame-works have been used to derive differential equation

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models that can approximate large-scale behaviormore efficiently or infer parameters for large-scalebehavior models.

Inherent to the presence of different spatialand time scales within biological systems, anothersignificant distinction occurs between data from invitro and in vivo experiments. A large amount ofbiological data, particularly at the molecular level,is obtained from in vitro experiments. Due to ex-perimental complexity, observations are often timesrestricted to single spatial and/or temporal scales.As a result, observations from in vitro and in vivoexperiments can have very different outcomes. In thecase of prion disease, aggregates within an organismare fragmented by chaperones, protein degradationfactors are present and protein is continually beingsynthesized. However, in in vitro assays, fragmenta-tion necessarily operates without the complete cellu-lar machinery and is typically operating under verydifferent concentrations. Thus, building a modelthat can resolve the discrepancy between in vitroand in vivo experiments by encompassing in vivoprocesses that may be missing at the in vitro scalecan prove very helpful. Mathematical and compu-tational models are uniquely positioned to capturethe connectivity between these divergent scales ofbiological function, and have the potential to bridgethe gap between isolated in vitro experiments at themost basic scale and whole organism in vivomodelswith organism or population level output.

Choosing a Mathematical Framework—There are many different mathematical tools that canbe used to convert the interactions at each scale givenby the conceptual model into a mathematical frame-work (20, 70). One main distinction in mathemati-cal frameworks is deterministic versus stochastic. Inthe previous section (see Section 3) we considereda deterministic model of prion aggregation. That is,given an identical set of initial conditions, the modelwill always produce the same output. This is incontrast to a stochastic model, in which we considerthe evolution of the probability the system occupiesany particular state. Intriguingly, the same set ofbiochemical equations (see Figure 4 (Middle)) canbe translated into either a stochastic or deterministicframework with the Law of Mass Action. (See forexample, (17, 18).) Ultimately the scientific ques-tion will determine the precise mathematical frame-

work to be used. For example, if we arewe interestedin the probability of the appearance of a prion phe-notype, we need to consider a stochastic model ashas been done by many (see for example (71, 72)).

In addition to deterministic vs. stochasticframeworks, there are many different options for thetypes of equations that can be used inside each typeof model. For example, as detailed in Section 3, sys-tems of differential equations using the law of massaction kinetics can be leveraged to represent chem-ical reactions. In addition, systems of differentialequations (ordinary or partial) are also ideal for de-scribing the concentrations of signalingmolecules inboth intracellular (inside one cell) and extracellular(moving throughout many cells) domains.

Another choice to be made is whether tobuild a continuous or discrete mathematical frame-work. In many applications, one multiscale phe-nomenonwewish tomodel is how the interaction be-tween intracellular processes, such as protein aggre-gation, happening inside each individual cell withina tissue or colony, and the individual cell behav-iors lead to emergent patterns or phenotypes at theorganismal or population level. In such cases, itis helpful to treat the intracellular process happen-ing inside each cell as a continuous variable usingdifferential equations and model individual cells asdiscrete entities that interact. Models that representcells as discrete entities are generally referred to asagent-based or cell-based models, and this class ofmodel has been used in many different applications.(For reviews see (60–64, 64, 65, 65–67, 73–78).)

CONCLUSIONSThere has never been a better time for in-

terdisciplinary research between experimental andmathematical scientists. Although communicatingacross domains is still challenging, the potential ben-efit from using mathematical models in new settingsis significant. Moreover, with increasing availabilityof user friendly software it is easier for scientists ofall backgrounds to develop and analyze their ownmathematical models (79–83). It is our hope thatthe three-step pipeline we have identified will enablereaders to pursue developing their ownmathematicalmodels and spur productive discussions with math-ematical scientists. Finally, we recognize that de-veloping a new mathematical model is always chal-

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lenging and, for brevity, our review contains onlyone modeling scenario. We encourage interestedreaders to refer to the many textbooks that we have

found valuable in our own learning and teaching(16–22) or to one of the other reviews in buildingmathematical models (12–15).

Acknowledgments: This research was sponsored in part by the Joint DMS/NIGMS Initiative to SupportResearch at the Interface of the Biological and Mathematical Sciences (R01-GM126548).

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FIGURES

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Figure 1: Interplay between Experiments and Mathematical Models. Mathematical models can be usedto test hypotheses, probe changes in parameters, generate predictions and design new experiments.

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Figure 2: Building a Model of Enzyme Substrate Kinetics. (Left) A conceptual diagram illustrating thekey biochemical species important in the system along with their interactions. (Middle) Explicit descriptionof the biochemical reactions represented from the diagram. (Right) A mathematical model, a system ofordinary differential equations (ODEs), describing the rate of change of each biochemical species.

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Figure 3: The Law of Mass Action. The relationship of the rate of a reaction to the concentrations of thechemical species that are involved in the reaction is given by the law of mass action. This law states that therate of a reaction is proportional to the product of the concentrations of the reactants.

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Figure 4: Diagram to Differential Equations. (Left) A diagram depicting the key players (monomer =circle, aggregate = blocks) and their interactions. (Middle) The set of biochemical equations which governthe interactions between key biochemical players monomer = X and aggregates of size i = Yi. (Note thatbecause of the assumption of a minimum stable nucleus size of n0, our form of the fragmentation equationdepicted is correct only if i ≥ n0, j ≥ n0.) (Right) Differential equation model schematic depicting thetemporal evolution of the concentration of monomer (x(t)) and aggregates of each size i (yi(t)). See text formore information.

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Figure 5: Temporal Kinetics of Prion Aggregation. We plot the temporal evolution of the concentration ofmonomer x(t) and protein in aggregates Z(t) (Left) and the number of aggregatesY (t) (Right) for “theoretical”choice of parameters. We note that even though prion aggregates are present at the initial condition, there isa long delay before we see a significant decrease in the healthy protein. In (B), the conversion rate is doubledcausing the time it takes for the system to reach steady-state to be cut in half.

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Figure 6: Multiscale Nature of Biological Systems. A digram depicting the multiscale nature of priondisease dynamics. (A) Biochemical Scale. The key biochemical players and their interactions. Protein issynthesized at a rate α, monomers are converted into aggregates at a rate β and fragmentation of aggregatesoccurs at a rate γ. (B) Subcellular Scale. The process of conversion, fragmentation, synthesis anddegradation of protein aggregates occurs inside individual cells. The presence of molecular chaperones,protein degradation factors and other substances varies inside each individual cell and can impact the rateof biochemical interactions. (C) Multicellular Scale. Division in yeast occurs through budding, a processduring which protein and aggregates are segregated between the mother and daughter cell. Modelingindividual cell behaviors makes it possible to study the interplay of molecular, subcellular and multicellularphenomenon. (D) Colony/Tissue Scale. At the largest scale, prion disease in yeast manifests as differentphenotypes at the colony level (disease =white and disease free = red), themost interesting ofwhich is sectoredcolonieswhere sections of cells within amajority diseased colony lose all their aggregates and become diseasefree (84). Moreover, multiscale models hope to connect important components at the biochemical scale withobserved spatiotemporal patterns in colony phenotypes. In mammalian neurodegenerative diseases, diseasephenotypes are observed at the level of an organ (85–87).

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Mikahl Banwarth-Kuhn and Suzanne S. SindiHow and Why to Build a Mathematical Model: A Case Study Using Prion Aggregation

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How and Why to Build a Mathematical Model: A Case Study Using … · 2020. 1. 31. · HowandWhytoBuildaMathematicalModel canbeusedtodevelopamathematicalmodel. We then demonstrate