: Liao D, Tlsty TD. 2014
Evolutionary game theory for physical and
biological scientists. I. Training and validating
population dynamics equations. Interface Focus
4: 20140037.
http://dx.doi.org/10.1098/rsfs.2014.0037

# Annotation of game theory for scientists 1 and 2

## Abstract

- Summary: Evolutionary dynamics and game theory are paramount to understanding complex biological systems. In this paper, they present a method that will use evolutionary game theory and game theoretic dynamics equations to design treatment strategies for a biological problem.

## Introduction

- Experiment: Will use theoretic replicator equations to analyse population dynamics based on mehtod of intial rates.

- INITIAL RATES = The method of initial rates is a commonly used technique for deriving rate laws. As the name implies, the method involves measuring the initial rate of a reaction. The measurement is repeated for several sets of initial concentration conditions to see how the reaction rate varies.

- MATHMATICAL MODELING = "we define mathematical modelling as the development of a consistent set of physical propositions (assumptions), quantitative relationships (equations, qualitative shapes of function plots, etc.) and observations."

- Section 2 provides background to describe the need of knowledge on this subject and set up this tutorial.

- Section 3 shows how data and equations can be compared using paramter training and model validation (excited for section 3!)

- Section 4 provides a potential clinical impact of what was learned in section 3.

## Section 2: Need for a tutorial in game theoretic analysis of cell population dynamics

- At interfaces connecting the physical and biological sciences, we are limiting ourselves when trying to understand complex biological systems by using reductionist reasoning.
- Reductionist reasoning is like focusing on only a small number of details in a bigger picture.

- EGT (evolutionary game theory) = "models of cell interactions in which net expansion rates for cell subpopulations depend on the frequency with which different cell types are represented in the overall population."
- model of the strategies of the different cell types and how it affects there population.

- In the 1st reference, the paper talks about how EGT can be used to fight cancer. The general approach in fighting cancer is to target and eliminate one type of a set of three cell types. This ends in a relapse but if we can change the dynamics we can have the other two cell types outcompete the bad cell type.

## Section 3: Phase portrait comparison of differential equations and population dynamics

- This section will explain how EGT replicator equations can analyze an ecology of two populations.
- Progress --> train differential equations with paramaters --> validate the equations with additional unoriginal data --> use the equations to make predictions.
- Experiment
- proportions of x cells (px = x/(x+y)) and the proprotion of y cells (py = y/(x+y)) are grown in cultures over time and monitored.
- Three populations.
- One initial population of nearly unity for x cells and 0 for y cells (A)
- x increases with a factor of 3 * its orginal value.
- One initial populations with equal proportions (px = 0.5, py = 0.5) (B)
- x increases by a factor of 2 * its orginal value.
- One inital population of nearly unity of y cells and nearly zero of y cells.(C)
- x does not increase.
- differential equations containing this realtionship
- dx/dt = (Ap~x~ + Bp~y~)x
- dy/dt = (Cp~x~ + Dp~y~)y
- Instantaneous rate of change of population (dx) over time (dt) equal to a rate coefficent (Ap~x~ + Bp~y~) times the population size of x.
- Instantaneous rate of change of population (dy) over time (dt) equal to a rate coefficent (Cp~x~ + Dp~y~) times the population size of y.
- The parameters are A,B,C,D which will be explained in the next section
- Point of confusion: equations describe the dynamics of absloute population **numbers**, but ecological and evolutionary scientist write down governing equations for the dynamics of population **fractions**.
- This is done in a form which can be directly derived from the equations used in the paper or using an alternate measure of time.
- population dynamics requires less equations and parameters as the equations can be written as one equation with only two parameters
- The cost of this simplicity: the analysis of fractions might predict a decrease in the fraction of a relatively aggressive cell type, but if the overalll population increases over the same time period , the absolute size of the aggressive population might also have increased leading to disease progression.
- Thus, fractions show proportions but if all populations increased in size with the aggressive type just growing the slowest the aggressive will look like its shrinking but actually its still growing.

### 3.1 Training

- To train the parameters for the two equations we have to use the method of initial rates.
- individual equation parameters can be isolated by considering how population sizes vary while one subpopulation dominates (main idea). This is because to isolate a parameter you need unity of one population and almost zero for the other.

- **Population A** with unity of x (10,000 cells) and 100 cells of y.
- Since x is nearly unity and y is nearly zero the equation becomes: dx/dt = Ax
- this is a proportionality between the instantaneous rate of change of population x nad the population size of x.
- Implies that 1 * change of x / x * change of t = A (equation to find parameter)
- change of population size of x / population size of x * change in time = parameter A
- The slope of the line tangent to the earliest data points covers a rise of change of x = +5000 cells over the course of two days (change of t = 2).
- initial population size of x is 10,000
- subsituting into equation:
- **A** = 1 + 5000 cells / 10000 cells * 2 days = **0.25 d^-1^** (x cells for population A)
- Thus, using the same reasoning we can find parameter C (y cells for population A).
- **C** = 1 + 100 cells/ 100 cells * 2 days = **0.5 d^-1^** with a change of +100 cells over 2 days and an initial population of 100 cells.

- population B can be used to find the remainding parameters B and D.
- **B** = 1 - 100 cells/ 100 cells * 2 days = **-0.5 d^-1^** with a change of -100 cells over 2 days and an initial population of 100 cells. (x cells for population B)
- **D** = 1 - 5000 cells/ 100000 cells * 2 days = **-0.25 d^-1^** with a change of -5000 cells over 2 days and an initial population of 10000 cells. (y cells for population B)

- Next step is to make a velocity plot to represent our trained differential equations.
- First we make a hypothetical population of 1000 x cells and 1000 y cells.
- We estimate the change in population x accrued over a short interval of time with the equation: change of x = dx/dt (first equation on paper) * change of time = (Ap~x~ + Bp~y~) * x * change of time
- now we subsitute in our parameters, our initial proportions, our initial population size (1000 cells), and our time (1 day).
- change of x = ((0.25/day)*(0.5) + (-0.5/day) * (0.5)) * 1000 cells * 1 day = -125 cells
- this leaves us with a decrease of 125 cells from 1000 x cells to 875 x cells.
- Now using the same reasoning as we did to find the change of x cells we can find the y cells.
- change of y = ((0.5/day)*(0.5) + (-0.25/day) * (0.5)) * 1000 cells * 1 day = 125 cells
- this leaves us with an increase of 125 cells from 1000 y cells to 1250 y cells.
- Now we can plot the inital population (1000, 1000) and our final population of (875, 1250) with x-axis as x cells and y-axis as y cells.
- The arrow of the plot represents population over the course of one day. When you calculate many of these arrows you can group them on what you call a phase portrait or velocity plot.
- You create these plots by having a steep, intermediate, and shallow sloped line on the plot. Then you can calculate mulitple arrows on this line by just replacing the intial population values.

### 3.2 Validation

- Now we compare the phase portrait to an additional dataset that explores a different set of population compostion
- the purpose to to assess agreement in terms of magnitude and direction in our model.
- take a look at the velocity plot (h) for population g at point i and the velocity arrow corresponding to it.
- The arrow is roughly in the same direction as the line segment connecting i and j which shows direction and if you divide the line segment by 3 for 3 days the line segment lengths are the same (magnitude)

### 3.3 Prediction

- In the prediction section, they seem to validate the equations more by predicting if different intial values will follow the same predicted pattern as the velocity plot shows. If not, then teh eqautions fail.

## Section 4: Discussion

- This paper has shown how replicator dynamic equations can be trained and validated using population sizes measured in a co-culture over time.

### 4.1 Clinical implications

- We can use this data and technique when studying cancer.
- If we use one treatement and the population of x increases while y decreases their is a disease. Just as if a treatment increased y and decreased x.
- Then one would say to use a combination of the two drugs but if one notices the velocity plot. As populations grow in time the magnitude of the velocity population arrows increase showing an increase of both types of cells.
- Thus, would one notice this and just have to increase the dosage as one goes on?

## Supporting papers:

- 1. **Cancer phenotype as the outcome of an evolutionary game between normal and malignant cells**("https://www.nature.com/articles/6605288")
- continuous-time replicator equations to study the dynamics and equilibria of mixed populations of MM, osteoclast (OC) and osteoblast cells.

- 2. **Evolutionary stable strategies and game dynamics**("https://www.sciencedirect.com/science/article/abs/pii/0025556478900779?via%3Dihub")
- absolute population fractions instead of numbers in replicator equations. (See section 3)

---------------------------------------------------
: Liao D, Tlsty TD. 2014
Evolutionary game theory for physical
and biological scientists. II. Population
dynamics equations can be associated with
interpretations. Interface Focus 4: 20140038.
http://dx.doi.org/10.1098/rsfs.2014.0038

# Evolutionary game theory for physical and biological scientist 2

## Abstract

- summary: The second paper in a two part series that explains how mathmatical equations are used to analyze populations dynamics in biological systems like cancer. Try to associate interpretations with mathmatical derivation.

## 1. Introduction

- In the first manuscript they used differential equations to analyze a time course system of a popultion of cells.
- describing the dynamics of interacting populaions

- The word model refers to many things.
- statistican: parametrized equation
- in this situation the differetial equations could act like a model but for a physicist the term model refers to a set of physical propositions.
- for a physicist, modeling is more than writing down an equation that imitates a plot of data like a statiscian
- in the first paper they just addressed the process of training and validating equations using data but in this paper they woll associate differential equations and midels using mathmatical derivations. **So how can equations be associated with interpretations ?** (main question to be answered)

### 1.1 Manuscript organization

- Section 2 = common way evolutionary game theory is misinterpreted
- EGT is used in both economics and biology.
- Section 3 = show how EGT models can be properly associated with microscopic hypotheses using mathmatical derivations.
- Section 4 = "outline how choices of models, equations (or simulations) and experiments can be refined by considering biological knowledge of cell-cell interaction process, analysis and computer simulation techniques from the physical sciences, and avalibility of cell culture, manipulation, and imaging techniques from biology, physics and engineering."
- Discussion = adavancing the application of game theory in biology and our understanding of biological systems.

## 2. Evolutionary game theory does not require that cells display sophisticated intelligence

- The misinterpretation comes from shared terminology and use of game theory in economics and biological applications.
- For the purposes of the paper, game theory broadly encompases the reasoning and conclusions that follow from investigating how the outcome (ability to reproduce, paycheck, jail time) of an individual agent (cell, business person, criminal, etc.) depends on the status of other agents with the individual interacts.
- Game theory also includes the investigating how an agent's actions can be influenced by social context.
- Table one shows the comparison between comparative statics (economics) and evolutionary dynanics (biology)
- Cells don't need to make rational decisons because they explain a phenotype (strategy) that will determine if they reproduce or not. Thus, a population consists of many different phenotypes and, through the generations, genetic pruning will occur.
- This is known as EGT and the individual cells are replicators with the parameters formatted in payoff matricies.
- Solutions are abstract mathmatical relationships.

## 3. Mathematical derivation of replicator equations

- provide five examples of how EGT population dynamics equations can be obtained by quantitatively expressing assumptions about microscopic cell-cell interaction processes.
- First three examples show how equations can be obtained by proposing that true-breeding is triggered by cell-cell collisions, by proposing that cell type conversion is triggered by cell -cell collisons or by proposing that cells modulate their proliferation rates by taking census of their environment.
- The last two examples show how a general version of the equations at the start of the paper cna be obtained by assuming that more than two cells can collide or by assuming that cell surface receptors interactions display cooperativity.

### 3.1 Cell replication triggered by pariwise cell-cell collisions

- True breeding = parents produce offspring of same phenotype.
- first example shows how the orginal equations can be obtained by proposing that cells trigger each other to proliferate through pairwise collisons.
- In figure 2, (a) is tracking the trajectory of a x cell while (d) is tracking the trajectory of a y cell (focal individuals).
- Must calculate the number of progeny produced by each of these focal individuals.
- The soultion is mixed so the individuals will encounter a fix number of cells and we will label this rate 'r'.
- over a time interval (change in time), each cell collides with r * change in time other cells. In a fraction of these collisons, the focal will encounter an x cell and in the rest of the fraction a y cell.
- We propose that the solution is so well mixed that the fractions equal the population fractions or proportions p~x~ and p~y~.
- Also each collison causes a replication of the focal.
- A collison between an x cell focal and an x cell triggers the focal to produce A/r additional x cells.
- In the figure A/r = 3 cells.
- The collison at c is between an x focal and a y which contributes B/r cells.
- B/r in the figure is 2 cells.
- This leaves us with the equation. (r* change in time) * p~x~ * A/r + (r * change in time) * p~y~ * B/r = change of x from focal indiviudal.
- Now because the focal is an x cell, we multiply the quantity given to us by the previous equation by the population of x cells anddivide out the time interval. This gives us a number consistant with our original equation (1.1).
- 1.2 can be derived the same way.
- We have derived this equation and interpretation but this is only one of many that can be true. More in next section.
### 3.2 Horizontal gene transfer triggered by cell-cell collisons

- showing how dynamics of cells that don't breed true are consitant with equations 1.1 and 1.2.
- Re-express equation 1.1 and 1.2 with different parameter names.
- dx/dt = (Ap~x~ + vp~y~) (3.3)
- dy/dt = (yp~x~ + Dp~y~) (3.4)
- Introduce a constant: þ
- B = v - þ
- C = y + þ
**THIS IS WHERE I GET STUCK. NEED A WALK THROUGH OF THIS MATHMATICAL DERIVATION**
- dx/dt (now equals) = (Ap~x~ + Bp~y~)x + þp~y~x
- dy/dt (now equals) = (Cp~x~ + Dp~y~)y + þp~x~y
- Let þ = +k~1~ - k~2~ and observe that p~x~y = p~y~x we finally get:
- dx/dt = (Ap~x~ + Bp~y~)x + k~1~p~x~y - k~2~p~y~x
- dy/dt = (Cp~x~ + Dp~y~)y + k~1~p~x~y + k~2~p~y~x

- Interpreting the equation:
- k~1~p~x~y can be interpreted as a rate at which cells of type y become cells of type x. This rate increases in proportion to the number of cells of type y available to become cells of type x.
- This proportionality coefficent has a constant factor k~1~ and a copy of p~x~, showing that the frequency of this process increases with increasing the proportion of type x in the environment.

- This is not true breeding because when an x cell collides with a y cell the x cell can release an exosome that integrates into the genome of the y cell.
- the term k~2~p~y~x is the same thing as the previous term but in the opposite direction or x cells becomeing y cells because a y cell gave an exosome to an x cell causing it to change phenotype.
- could also create a new third cell which colliding which is observed in metastaic mammary cells.

### 3.3 Autocrine and paracrine signalling

### 3.4 N-way cell-cell collisons dynamics

### 3.5 Cooperative binding of signal factors

## 4. Designing studies of population interactions

### 4.1 Biological knowledge

### 4.2 Techniques in mathmatical and computational modeling

#### 4.2.1 Structured populations

#### 4.2.2 Agent-based (individual-based) models

#### 4.2.3 Stochasticity

### 4.3. Experimental systems

### 5. Discussion

### 5.1 Engineering clinival and in vivo monitoring

### 5.2 Beyond a focus on the cell

### 5.3 Developing a fundamental physics of living systems

## Supporting papers:

- 1. **Spontaneous fusion between metastatic mammary tumor subpopulations**("https://onlinelibrary.wiley.com/doi/10.1002/jcb.240360204")
- cells collide and create a third type of cell different from the two colliding.