*in vitro*experiments with results from

*in vivo*and whole-organism studies. However, developing useful mathematical models is challenging because of the often different domains of knowledge required in both math and biology. In this work, we endeavor to provide a useful guide for researchers interested in incorporating mathematical modeling into their scientific process. We advocate for the use of conceptual diagrams as a starting place to anchor researchers from both domains. These diagrams are useful for simplifying the biological process in question and distinguishing the essential components. Not only do they serve as the basis for developing a variety of mathematical models, but they ensure that any mathematical formulation of the biological system is led primarily by scientific questions. We provide a specific example of this process from our own work in studying prion aggregation to show the power of mathematical models to synergistically interact with experiments and push forward biological understanding. Choosing the most suitable model also depends on many different factors, and we consider how to make these choices based on different scales of biological organization and available data. We close by discussing the many opportunities that abound for both experimentalists and modelers to take advantage of collaborative work in this field.

## Introduction

## What does it mean to model something?

*i.e.*different initial conditions and/or parameters) and observe whether or not the predicted outcome occurs. The mathematical model can be used to study how sensitive one output of interest is to increasing or decreasing the amount of other factors. The mathematical model can also be used to aid in the design of new experiments. However, this process is not one-sided. Developing a useful mathematical model relies on having a solid understanding of the system under study. Unknown parameters may be “fit” by comparing model output with data, and the model itself can be “validated” by predicting an unexpected experimental outcome.

*S*reacting with an enzyme

*E*to form an enzyme substrate complex

*E*:

*S*, which is converted into a product

*P*and the enzyme. The diagram for this system is given in Fig. 2 (

*left*). Note that this diagram is by nature conceptual; we are typically specifying only the interactions and not the exact quantitative nature of those interactions. We emphasize that the creation of this type of diagram is particularly important in interdisciplinary work because its visual nature makes it accessible to all researchers and provides a common ground for moving forward.

*middle*). In our experience studying biological systems at the molecular scale, biochemical equations provide a natural way to list these interactions. (For a discussion of how this step is handled at different biological scales, see Section 4.) It is also at this step that decisions about stoichiometry and the form of interactions are clarified (it is possible that complex interactions are simplified and multistep reactions determined) and rates (more so as symbols than numerical values) are assigned to specific reactions. Most importantly, this step requires gathering knowledge from experiments and experts in the field to incorporate what is currently understood about the system of interest.

*E*) and substrate (

*S*) form a complex (

*E*:

*S*) at rate

*k*

_{1}but that this complex itself is reversible at rate

*k*

_{−1}. This complex may also, at rate

*k*

_{2}, result in the creation of a product (

*P*) and a return of enzyme in the complex to the free pool. (In this example, the system as defined does not include synthesis or degradation, so mass is preserved. This simplification is reflected in our biochemical equations because there is no synthesis or degradation and we only show three reaction rates.)

*i.e.*integer, real number, Boolean, etc.), and why the choice of representation is appropriate for each component. (For simplicity and because of their popularity, in this paper, we will consider a deterministic model of concentrations. We discuss other possibilities in Section 4.)

*i.e.*proportional) in terms of measurable quantities (

*i.e.*concentration of the reactants). In our example, the biochemical reactions lead to the well-studied system of differential equations depicted in Fig. 2 (

*right*). (We note that although the law of mass action was developed for molecular processes, it has been used as a model for interactions at larger scales and indeed is the foundation for many mathematical models in ecology and epidemiology (, ,

*i.e.*usually only the product concentration

*P*(

*t*) can be visualized). However, the mathematical model can reveal the “hidden” concentrations of all other variables too. Because we suspect most readers are familiar with this simple system, in the next section we explore developing a mathematical model in a more complicated setting and demonstrate how analytical and numerical methods are used to study the model.

## Case study: Protein aggregation kinetics

*Saccharomyces cerevisiae*has several harmless phenotypes that we now know to be caused by prion proteins (

*n*

_{0}, of misfolded protein monomers (

*PSI*

^{+}] prion in

*S. cerevisiae*, the nucleus is thought to consist of five misfolded monomers (

*n*

_{0}in size are highly unstable and are thought to rapidly resolve into monomers (

### Setting the stage for a mathematical model

*n*

_{0}.

*n*

_{0}in our model will be considered not stable and quickly resolved into healthy monomer. As we said before, once the species and interactions are identified, it is advisable to codify these properties in a picture or diagram. In addition to being a useful tool for a mathematical modeler on its own, the visual medium of pictures facilitates communication between researchers in different domains. Agreeing on a diagram of interactions is an important first step in mathematical modeling.

*left*). We consider two types of biochemical species: soluble (monomer) and aggregates. We have also introduced the idea that the number of blocks in an aggregate indicates its size (

*i.e.*conversion shows an aggregate of size 3 becoming an aggregate of size 4). We have also diagramed the basic nature of the conversion and fragmentation reactions. Notice that in drawing our fragmentation equation, we had to make a decision about what happens when fragmentation creates an aggregate with a size below that of the nucleus (

*n*

_{0}= 2). Whereas one choice could be that those proteins completely degrade, we have instead decided to have these proteins return to the soluble monomer state. (This is consistent with the NPM (

*in vivo*prion kinetics, we assume the protein synthesis and degradation machinery to be functioning. Of course, aggregation models of

*in vitro*aggregation may elect to consider different processes, as in Ref.

*middle*)). The

*left-hand side*of the biochemical equations gives the reactants, what is needed or consumed for the reaction to be carried out, and the

*right-hand side*shows the resulting products. The reaction rate designates the speed at which the reaction takes place. The units of the reaction rate typically depend on the order (number of reactants) of the reaction itself. (For example, in a first-order reaction (only a single reactant), the reaction rate has units of (time)

^{−1}.)

*middle*).

*in vitro*experiments, the best fitting mathematical model was one in which the middle of amyloid fibers was more likely to fragment than the ends (

*i*+

*j*), there are two fragmentation sites that would result in two aggregates with distinct sizes (

*i*and

*j*). (Note that if

*i*=

*j*, then although there is only one site that could create these aggregates, the stoichiometry remains the same, because we produce two aggregates of the same size.) Similar biochemical reactions govern the processes of synthesis (α) and degradation (μ). (Note that in the original NPM, distinct degradation rates were considered for monomers and aggregates. However, for simplicity, we consider both processes to occur at the same rate μ.)

### Development and analysis of the nucleated polymerization model: A mathematical model of prion aggregation

- •Can we determine what causes prion aggregates to be cleared by manipulating biochemical rates?
- •Why is there a long incubation time for many prion diseases?

*right*)). In other words, we define

*x*(

*t*) to be the concentration of healthy monomer and

*yi*(

*t*) to be the concentration of an aggregate of size

*i*and track their evolving concentrations in time. We use the law of mass action to convert our biochemical equations into differential equations. As mentioned above, this law states that the instantaneous rate of a reaction is a product of a reaction rate and the product of the concentration of all reactants at that time. Let's look at an example. Suppose the current concentrations of the monomer

*x*and aggregates

*yi*are known. What is the instantaneous rate of aggregate conversion of monomer by an aggregate of size

*i*?

*top*biochemical equation in Fig. 4 (

*middle*), this rate is given by 2β times the concentration of monomer

*x*times the concentration aggregates of size

*i*,

*yi*,

*n*

_{0}, our system formally has infinitely many reactions to consider! We will soon see how this is no obstacle for mathematical modeling.) We then combine these reaction rates into a differential equation for each biochemical species. The differential equations are themselves simply the sum of reactions that create the given species (rate in) minus the reactions that consume the species (rate out). Let's consider the differential equation for the monomer concentration

*x*(

*t*),

*x*(

*t*). Second, monomer is consumed during conversion by every possible aggregate size. We have already calculated the rate of conversion by an aggregate of size

*i*in Equation 1. In calculating this bulk rate, we need to sum the rate of conversion over all possible aggregate sizes,

*x*(

*t*). As for the “rate out” process, there are two types of reactions that create monomer. First, monomer is synthesized at rate α. Second, monomer is created when aggregates are fragmented in such a way that one piece of the recently fragmented aggregate is below the minimum stable size.

*n*

_{0}. This aggregate has

*n*

_{0}− 1 fragmentation sites, and every possible fragmentation event will result in the creation of two aggregates with size less than

*n*

_{0}. Due to our assumption that any aggregate below the nucleus size

*n*

_{0}is not stable, each of the fragmented pieces will be resolved into

*n*

_{0}monomers. Thus, the rate of monomer creation by fragmentation of aggregates of size

*n*

_{0}is given by the following,

*n*

_{0}+ 1) there are

*n*

_{0}fragmentation junctions. Two of them will create an aggregate of size

*n*

_{0}and recover a single monomer, and the remaining (

*n*

_{0}− 2) junctions would result in all (

*n*

_{0}+ 1) proteins in the aggregate returning to the monomer state. (For example, suppose

*n*

_{0}= 3 and an aggregate of size 4 is going to be fragmented. There are three fragmentation sites, each of which is equally likely to be chosen. If the first or third fragmentation site is chosen, the resulting pieces will be size 1 and size 3. Because

*n*

_{0}= 2, the size 1 piece will return to the monomer state, and the size 3 piece will remain an aggregate. If the second fragmentation site is chosen, the resulting pieces will both be 2, which is smaller than

*n*

_{0}, and the result will be 4 monomers.) Thus, the rate of monomer creation by fragmentation is given by the following,

*n*

_{0}(

*n*

_{0}− 1) as before. In fact, this pattern holds for all aggregate sizes but requires slightly different reasoning when the aggregate size exceeds 2

*n*

_{0}. In this case, at most, one of the pieces resulting from a fragmentation event will be smaller than

*n*

_{0}. (For example, suppose

*n*

_{0}= 3 and an aggregate of size 6 is going to be fragmented. There are five sites, each of which is equally likely to be chosen. If the middle fragmentation site is chosen, the resulting pieces will each be 3, the same as

*n*

_{0}. Every other fragmentation site will result in one piece that is smaller than

*n*

_{0}, which will return to the monomer state, and one piece larger than

*n*

_{0}, which will remain an aggregate. The two outermost fragmentation sites will result in recovering one monomer, and the last two will recover two monomers. The sum of monomers recovered, regardless of the original aggregate size, will always be

*n*

_{0}(

*n*

_{0}− 1).

*x*(

*t*) is as follows,

*x*(

*t*) as follows,

*yn*

_{0}(

*t*) and

*yi*(

*t*) when

*i*>

*n*

_{0},

*n*

_{0}). We will consider each approach below to answer the two questions we posed originally.

### Stability of the prion aggregate system: Analytical approach

*yi*, we will consider two new quantities about the aggregates as follows:

and

*Y*(

*t*) represents the total concentration of prion aggregates, whereas

*Z*(

*t*) represents the concentration of total protein in prion aggregates. Remarkably, our system of differential equations in

*yi*can be arranged into a set of differential equations for

*Y*(

*t*),

*Z*(

*t*), and

*x*(

*t*) (our original monomer population),

*x*,

*Y*, and

*Z*that satisfy the following,

*x*= α/μ,

*Y*=

*Z*= 0. In this case, all protein is in the healthy monomer state. The second nontrivial steady state (

*x**,

*Y**,

*Z**) is more clearly shown in the following relationships,

where

*R*

_{0}value to represent the basic reproductive number (

*R*

_{0}represents the number of secondary infections produced by one primary infection in a susceptible population. When

*R*

_{0}< 1, no matter how many aggregates are initially present, they will all eventually be cleared, and the system will converge to the aggregate-free steady state. When

*R*

_{0}> 1, the introduction of any initial amount of aggregate will lead the system to converge to the nontrivial steady state.

*R*

_{0}changes from a value greater than 1 to a value smaller than 1, prion aggregates will be completely cleared. But what parameter values should we change in order to arrive at such an

*R*

_{0}value? And by how much? Whereas it might seem obvious that increasing the degradation rate μ above a certain threshold will cause prion aggregates to be cleared, lack of an algebraic expression for

*R*

_{0}would make it almost impossible to determine the precise degree of change in parameters to produce the desired output.

*A*produce an

*R*

_{0}value of ∼5.7. As expected, because this

*R*

_{0}value is greater than 1, prion aggregates do persist and reach a positive stable steady state as shown. However, if we fix all other parameters and only change the fragmentation rate γ, we can use Equation 19 above to show how big or small γ must be to produce an

*R*

_{0}value of <1 where prion aggregates are cleared. Keeping all parameters the same as in Fig. 5

*A*, we have the following,

### Dynamics of the prion aggregate system: Numerical approach

*x*(

*t*) and

*Z*(

*t*) (

*A*and

*B*,

*left*) as well as the aggregate concentration

*Y*(

*t*) (

*A*and

*B*,

*right*) for a particular choice of biochemical parameters upon the introduction of a very small amount of prion aggregate

*Z*(0) =

*Y*(0) = 1 nm. (The biochemical parameters were chosen to roughly mirror the properties of the [PSI

^{+}] weak strain in yeast but plotted on a time scale relative to mammalian prion disease (

*A*, we see that the amount of healthy monomer

*x*(

*t*) decreases relatively slowly; after 400 weeks, only 9% of the protein has changed to the prion (misfolded) state. This is because, initially, the aggregates are at a very low concentration, and thus conversion of monomer is favored over fragmentation. Because conversion only adds to existing aggregates (but does not create them), this has only a modest impact on the pool of healthy protein. However, just because we cannot see much of a decrease in

*x*(

*t*) does not mean we are safe!

*B*, we doubled the conversion rate, and as a result, the system reached its steady state almost twice as fast! We next describe how this model was used by us and previous researchers to learn about prion aggregation processes.

### Interaction between mathematical models and experiments

*n*

_{0}) to the exponential growth rate in aggregated protein they observed during early phases of prion disease through the relationship of the parameters to the

*R*

_{0}(

*X*(0),

*Y*(0),

*Z*(0). Indeed, as protein aggregation modeling has advanced, easy-to-use computational pipelines have been developed for researchers interested in fitting their

*in vitro*aggregation curves (

*PSI*

^{+}] prion (

## Discussion

- •What is the catalytic rate of a particular enzyme reaction? To address this question, a model must minimally include processes at the molecular scale. However, depending on desired complexity, the choice could be made to model the reaction of interest as a one-step process or include several successive steps in the reaction. In addition, important steps impacting an enzyme reaction could occur at the same scale, or some step might occur on a larger scale, such as cell behaviors or nutrient gradients that impact the environment where the reaction is taking place.
- •Why does my experimental system have such high variance? Sensitivity analysis tools offer great methods to identify the cause of variance in complex biological systems. To answer this question, a model could consider one scale only and investigate the impact of variability of one or two factors at the same scale. Or the model could incorporate processes at the molecular scale and processes at the cell or tissue scale and use global sensitivity analysis methods to identify which factor(s) has the most impact on the overall variance of the system.
- •What are the best time points to sample my experimental system to observe dynamics of the behavior I wish to study? Biological processes in the same system happen at different time scales. For example, chemical reactions happen much faster than the time it takes for a cell to divide or move across an agar plate. Depending on the scientific question being asked, samples at different time points can provide necessary experimental data to calibrate model components at different scales.

### Incorporating processes at multiple biological scales

*i.e.*genetic mutations, formation of a blood clot, and even the growing and shortening phases of microtubules in individual cells) (

*i.e.*molecular biologists

*versus*cellular, organismic, or population, etc.). Mathematical and computational models provide a tool to integrate knowledge from different scales and identify how collective interactions on a fundamental scale can give rise to large-scale phenomena (Fig. 6). Thus, one challenge in building a mathematical model is determining on what scale (or scales) the key players and interactions occur. One important question to ask is which scale(s) of resolution will provide the most information for understanding the underlying mechanisms of the system? An equally important and converse question is whether there is a scale of resolution that offers no insight and should be simplified or ignored.

*A*). For this reason, the majority of mathematical models developed to answer the question of what causes prion aggregates to be cleared have only considered interactions between components on the molecular scale (see Section 3). However, in some instances, these models were not able to reproduce experimental data. In this case, the given modeling framework can be extended to include processes at the next level of organization in order to attempt to produce model results that agree with experimental data (Fig. 6

*B*).

*et al.*(

*C*). One of the most unique complications introduced by studying prion disease in yeast comes with the consideration that prions in yeast propagate within a colony of growing and dividing cells. The time it takes for protein to change configurations is much faster than the time it takes for a cell to grow or divide. Thus, an interesting question to consider is how cellular behaviors such as cell cycle length, protein segregation at the time of division, and/or age of cells impact protein aggregation dynamics throughout the entire colony (Fig. 6,

*B*and

*C*). To address this question, a new model must include interactions between intracellular components within each individual cell, individual cell behaviors impacting intracellular dynamics, and cell-cell interaction between many different cells in the same colony. One type of well-established modeling framework that is capable of integrating molecular, subcellular, and cellular level processes is agent-based models. Agent-based models represent cells as discrete units that interact with each other and can also carry out individual cell processes such as protein aggregation, division, and growth (for reviews, see Refs.

*D*). In many studies, population level data are the only quantitative output available from experiments. However, the concentration of a protein or other cellular constituent is known to vary considerably among cells in the same colony. This heterogeneity is thought to arise from several sources, including differences in kinetic rates between individual cells and distribution of cellular constituents at the time of cell division. Connecting interactions between components at the fundamental scale with population level phenotypes is experimentally challenging. However, in multiscale models, perturbations of parameters at the fundamental scale (

*i.e.*protein modifications) can generate observable and measurable changes to coarse-grained outputs at the population level (

*i.e.*colony phenotype). Integrating processes across different spatial and temporal scales can be achieved using agent-based models described earlier. Furthermore, agent-based models have been used in many applications, including tumor growth, blood clot formation, stem cell regulation in plants and understanding the interplay of biochemical and molecular parameters on individual cell behaviors. Another class of model that can be used to study dynamics of entire cell colonies, tissues, and organs are continuous models that use ordinary differential equations or partial differential equations to represent the growth of a tissue as one continuous sheet or the change in the shape or position over time of the edge of a colony or group of cells as one continuous boundary. In many cases, discrete, agent-based modeling frameworks have been used to derive differential equation models that can approximate large-scale behavior more efficiently or infer parameters for large-scale behavior models.

*in vitro*and

*in vivo*experiments. A large amount of biological data, particularly at the molecular level, is obtained from

*in vitro*experiments. Due to experimental complexity, observations are often restricted to single spatial and/or temporal scales. As a result, observations from

*in vitro*and

*in vivo*experiments can have very different outcomes. In the case of prion disease, aggregates within an organism are fragmented by chaperones, protein degradation factors are present, and protein is continually being synthesized. However, in

*in vitro*assays, fragmentation necessarily operates without the complete cellular machinery and is typically operating under very different concentrations. Thus, building a model that can resolve the discrepancy between

*in vitro*and

*in vivo*experiments by encompassing

*in vivo*processes that may be missing at the

*in vitro*scale can prove very helpful. Mathematical and computational models are uniquely positioned to capture the connectivity between these divergent scales of biological function and have the potential to bridge the gap between isolated

*in vitro*experiments at the most basic scale and whole organism

*in vivo*models with organism or population level output.

### Choosing a mathematical framework

*versus*stochastic. In Section 3, we considered a deterministic model of prion aggregation. In other words, given an identical set of initial conditions, the model will always produce the same output. This is in contrast to a stochastic model, in which we consider the evolution of the probability that the system occupies any particular state. Intriguingly, the same set of biochemical equations (see Fig. 4 (

*middle*)) can be translated into either a stochastic or deterministic framework with the law of mass action (

*e.g.*see Refs.

*e.g.*see Refs.

*versus*stochastic frameworks, there are many different options for the types of equations that can be used inside each type of model. For example, as detailed in Section 3, systems of differential equations using the law of mass action kinetics can be leveraged to represent chemical reactions. In addition, systems of differential equations (ordinary or partial) are also ideal for describing the concentrations of signaling molecules in both intracellular (inside one cell) and extracellular (moving throughout many cells) domains.

## Conclusions

- Kluyver T.
- Ragan-Kelley B.
- Pérez F.
- Granger B.
- Bussonnier M.
- Frederic J.
- Kelley K.
- Hamrick J.
- Grout J.
- Corlay S.
- Ivanov P.
- Avila D.
- Abdalla S.
- Willing C.

## Supplementary Material

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This work was supported in part by the Joint Division of Mathematical Sciences (DMS)/NIGMS, National Institutes of Health, Initiative to Support Research at the Interface of the Biological and Mathematical Sciences (Grant R01-GM126548). The authors declare that they have no conflicts of interest with the contents of this article. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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