Seminar in mathematical Biology Theoretical issues in modeling

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Seminar in mathematical Biology Theoretical issues in modeling Yoram Louzoun Nadav Shnerb Eldad Bettelheim

Seminar in mathematical Biology Theoretical issues in modeling Yoram Louzoun Nadav Shnerb Eldad Bettelheim Sorin Solomon

Overview 4 Dynamical systems are traditionally modeled using ODEs. 4 However most of the

Overview 4 Dynamical systems are traditionally modeled using ODEs. 4 However most of the assumptions under which ODEs are correct are not valid in biological systems. 4 We analyze some prototypic systems and present a new method to analyze the behavior of stochastic spatially extended systems. 4 Using our results, we show that well known concepts, such as Malthusian growth and ecological niche are the result of over simplifications.

Fisher Equation 4 The simplest system contains a single agent and three processes: Birth,

Fisher Equation 4 The simplest system contains a single agent and three processes: Birth, death and competition. 4 Such system is represented by a Fisher Equation. 4 is the birth rate- the death rate, is the competition rate

Fisher waves 4 This equation has two solution (0) and ( / ). The

Fisher waves 4 This equation has two solution (0) and ( / ). The first solution is stable for (0 ) and the second solution is stable for ( >0). 4 If ( >0) the stable solution will invade the unstable solution forming a Fisher front with a width advancing with a velocity 4 Small fluctuation from the stable steady state of wavelength k decay at as exp[- (| |+k 2)t]. Fisher R. A 1937

Catalytic noise 4 In many biological and especially in social systems the creation of

Catalytic noise 4 In many biological and especially in social systems the creation of a new substance is induced by a catalyst. 4 The catalyst can have a very low density. This low density induces fluctuation in its local density. 4 The fluctuations in the catalyst density affects the reaction rates of the reaction they catalyze.

Full system 4 In order to asses the effect of such noise we model

Full system 4 In order to asses the effect of such noise we model a spatially extended systems with a catalyst and a simple reaction. 4 We model an agent (B) proliferating in the presence of a catalyst (A) with a probability , and dying with a rate . 4 Both the catalyst and the agent are diffusing 4 We ignore at this stage the non linear term.

System description 4 A+B->A+B+B , 4 B-> , A + B B A B

System description 4 A+B->A+B+B , 4 B-> , A + B B A B B

Naive PDE 4 The PDE describing this system is: 4 If we let the

Naive PDE 4 The PDE describing this system is: 4 If we let the A diffuse for a long time, before B reactions starts the A distribution will be: 4 A(x)=A 0

Malthus 4 The solution to these equation is an extrapolation from Malthus: 4 c

Malthus 4 The solution to these equation is an extrapolation from Malthus: 4 c is a positive constant, and is 0 for the mode representing uniform initial conditions of B(x).

Naïve PDE II 4 The fastest growing mode has an exponent of: - 4

Naïve PDE II 4 The fastest growing mode has an exponent of: - 4 If - >0, the population will grow until saturated by non linear terms, while if - <0 the population will exponentially decrease to zero (since B is discrete).

Simulation 4 We simulate this system, with a lattice constant l containing As and

Simulation 4 We simulate this system, with a lattice constant l containing As and Bs. 4 Each point in the lattice can contain an infinite number of A and Bs. 4 Interaction are only within the same lattice point. The interaction rate is /ld 4 The diffusion probability from a lattice point to its neighbor is: D/l 2 4 We simulate a case when the death rate is much higher than the birth rate: (A /ld - ) < <0.

Simulation Results 4 Even for a very high death rate the total population increases

Simulation Results 4 Even for a very high death rate the total population increases indefinitely (A= 0. 1, /ld =2, =1, A =-0. 8).

Even Simpler Example I 4 In order to understand this strange result, lets simplify

Even Simpler Example I 4 In order to understand this strange result, lets simplify our system. We will make it one dimensional with 18 cells. 4 Assume an A concentration of A=0. 2, /l =1 and =0. 5.

Even Simpler Example II 4 At t=0 The Bs have a random distribution. 4

Even Simpler Example II 4 At t=0 The Bs have a random distribution. 4 At t=1 Half the B disappeared and 20% of new Bs were created. 4 At t=2 the same process takes place…. .

Even Simpler Example III 4 But this is not the social reality, the As

Even Simpler Example III 4 But this is not the social reality, the As are discrete entities. 4 At places where there are no As all the Bs will disappear. 4 At places where at least one A exists the Bs will prosper.

Basic Idea 4 This is the result of the combination of an exponential growth

Basic Idea 4 This is the result of the combination of an exponential growth and a linear cut-off 4 In an inhomogeneous situation, regions where the local birth rate is lower than the death rate, the total population will simply shrink to zero. 4 In regions where the local birth rate is higher than the death rate the total population will increase to infinity. 4 After a finite time, only regions with a high B population will influence the average B population.

Single A Approximation 4 If only a single A exists, then the average A

Single A Approximation 4 If only a single A exists, then the average A population is close to zero. 4 If the A is fixed in space then the B at the location of the A obeys:

But, What About Diffusion? ? ? 4 After a some time the A will

But, What About Diffusion? ? ? 4 After a some time the A will diffuse, and all this story will not work any more. 4 Yes!, but the B also diffuses, and by the time the A diffuses there will already be Bs at the new position for the island to keep growing. 4 In every jump the A performs it lands in a region occupied by Bs, but with a lower concentration then at the center of the island.

Diffusion Only Makes it More Interesting 4 The size of the B island is

Diffusion Only Makes it More Interesting 4 The size of the B island is growing linearly. 4 The As on the other hand diffuses, so that they move with a rate proportional to the square root of t.

Island Shape 4 The neighboring cells have a constant input from the cell containing

Island Shape 4 The neighboring cells have a constant input from the cell containing the A and the same diffusion and death rate:

A jumps 4 When A jumps to a neighboring cell, the Ln of the

A jumps 4 When A jumps to a neighboring cell, the Ln of the B population at the A position decreases by: 4 The population at the center of the island is thus:

Emerging Island

Emerging Island

Cooperative effects 4 We simulate a system with =0. 5, =1, and <A>=0. 5

Cooperative effects 4 We simulate a system with =0. 5, =1, and <A>=0. 5 4 ( <A>- )=-0. 75 4 Although in lattice points containing a single A the B population decreases. In points containing 3 or more As the B population increases. 4 If the distribution of As is poissonian, there will be a macroscopically large number of points with 3 or more As in them

Low A diffusion 4 The A concentration has a Poisson distribution. 4 For any

Low A diffusion 4 The A concentration has a Poisson distribution. 4 For any m and any value of <A>, if the volume of space is large enough there will be a point containing m A agents.

Low A diffusion 4 For any value of , and DB, there is a

Low A diffusion 4 For any value of , and DB, there is a value m that will obey: 4 The B population will grow precisely in these points. 4 In other words, in the low A diffusion regime the B population grows following the maximum of A and not its average.

Low A diffusion

Low A diffusion

Master Equation 4 The situation of the system can be described using a master

Master Equation 4 The situation of the system can be described using a master equation for the probability to have A 1 As and B 1 Bs in the first cell… P(A 1 B 1. . ANBN). 4 The master equation for a single point on the lattice is:

Translation to quantum formalism 4 In order to solve this equation, we replace the

Translation to quantum formalism 4 In order to solve this equation, we replace the classical probability function by a quantum wave function: 4 >= Pnm/n!/m!(a†)n(b †)m|0>. 4 We define creation and destruction of A and Bs using the quantum creation and annihilation operators: a, a†, b, b†. 4 Adding a new A particle is simply: a†| >, and destroying a particle is: a| >.

Creation and annihilation. 4 These operators follows the following simple rules: 4 a†|n, m>=

Creation and annihilation. 4 These operators follows the following simple rules: 4 a†|n, m>= |n+1, m>; a|n, m>= n|n-1, m> 4 b†|n, m>= |n, m+1>; b|n, m>= m|n, m-1> 4 [a, b]=0 ; [a, a†]= a a†- a† a =1 4 Counting particles in a given location is: 4 a†a|n, m>= n a†|n-1, m>=n|n, m>

Hamiltonian 4 We replace the Hamiltonian with a quantum Hamiltonian, and obtain a non

Hamiltonian 4 We replace the Hamiltonian with a quantum Hamiltonian, and obtain a non imaginary schroedinger equation. 4 ’=H

Mean field 4 We replace the operators by their vacum expectation values, scale the

Mean field 4 We replace the operators by their vacum expectation values, scale the system and replace the interaction with neighbors to a continous gradient of a and b.

RG -2 D

RG -2 D

RG 3 D 4 In 3 D on the other hand there is a

RG 3 D 4 In 3 D on the other hand there is a phase transition, and in some of phase space the ODEs are precise.

Local Competition 4 The Bs may compete over a local resource (food, space ,

Local Competition 4 The Bs may compete over a local resource (food, space , light…). This local competition is limited to Bs living on the same lattice site. 4 The local competition will mot change the overall dynamics, but it will limit the size and total population of each B island. 4 Any first order interaction can be described in the form of proliferation and death, while any second order mechanism can be described as a competition mechanism.

Simulation of competition 4

Simulation of competition 4

Global Competition 4 The B agents may also compete over a global resource. This

Global Competition 4 The B agents may also compete over a global resource. This happens if the radius in which B compete is larger then the interaction scale between Bs. 4 For example cells competing for a resource in the blood, animals competing over water, plankton competing over oxygen. 4 Large scale competition is described as an interaction with the total population over some scale The competition reaction is : 4 B+<B>-><B>

The world company 4 When the A diffusion rate is low only one island

The world company 4 When the A diffusion rate is low only one island of Bs is created around the maximal A concentration. The high B population in this island will inhibit the creation of any other B island. 4 When the As diffuse fast, a number of large B islands are created. These islands look for food (High A concentration ). They can split, merge or die. 4 The life-span of these islands is much larger than the lifespan of a single B. 4 These emerging islands will lead to the creation of intermittent fluctuations in the total B population.

Predator prey systems 4 We have shown that the classical PDE treatment of a

Predator prey systems 4 We have shown that the classical PDE treatment of a ver simple autocatalytic systems is wrong. 4 One of the reasons for the large difference is the high correlation between a and b fluctuations. 4 We will show that PDE fail in system with anti-correlation terms

Predator pray systems 4 Lets denote a pray by a, and a predator by

Predator pray systems 4 Lets denote a pray by a, and a predator by b. 4 The pray population grows, unless destroyed by the predator. 4 The predator population is growing when it is “eating” the pray. 4 The predator population is limited by death (linear) and competition (non-linear)

Ecological niches 4 The equations describing the system are: 4 These equations have 2

Ecological niches 4 The equations describing the system are: 4 These equations have 2 fix points: (0, 0) and 4 ([ / + ]/ , / ) , but only the non zero fix point is stable. 4 This is the origin of the ecological niche concept.

Infection Dynamics-ODE

Infection Dynamics-ODE

Infection Dynamics-ODE

Infection Dynamics-ODE

Infection 4 The mean field approximation of similar predator pray dynamics are used to

Infection 4 The mean field approximation of similar predator pray dynamics are used to describe infection dynamics, where the pathogen is the pray and the immune system cells are the predator. 4 As in the previous cases, the ODEs fail to take into account some elemental biological features which makes their results obsolete.

Missing elements. 4 The two main elements missing from the differential equations are: –

Missing elements. 4 The two main elements missing from the differential equations are: – The discreteness of the immune cells and pathogens. – The time required to produce an immune cell. – The saturation of immune cells reproduction capacity. 4 We are explicitly simulating an immune system, but these elements are present in every P-P system.

Simulations.

Simulations.

Spatial distribution. 4 The previous result can be obtained either from a simulation where

Spatial distribution. 4 The previous result can be obtained either from a simulation where every point in space have the same random initial distribution, or from a SDE. 4 One can ask what happens if the pathogen is presented in a single point of space.

Infection Dynamics Simulations.

Infection Dynamics Simulations.

Random Spatial structure 4 The immune system is dwelling in an Euclidean space. 4

Random Spatial structure 4 The immune system is dwelling in an Euclidean space. 4 A more realistic simulation should contain random neighbors.

Global destruction. 4 An even more interesting dynamics can take place if the predator

Global destruction. 4 An even more interesting dynamics can take place if the predator has a preying range much larger than the prey diffusion radius. 4 This situation is very frequent. For example lions and tigers have a very wide preying range compared to the grazing range of zebras or antilopes.

Infection Dynamics Simulations.

Infection Dynamics Simulations.

P-P delay 4 We have ignored up to now the explicit delay between the

P-P delay 4 We have ignored up to now the explicit delay between the activation of the predator, and its capacity to destroy the pathogen.

Summary 4 ODEs fail completely in describing autocatalytic systems. 4 The total population of

Summary 4 ODEs fail completely in describing autocatalytic systems. 4 The total population of an agent with a lower proliferation rate than death will increase, in contradiction with the homogenous description. 4 The chance of survival are much more important in 2 D than in 3 D. 4 Very simple dynamics can create emerging objects with a long lifespan. In our case these objects are islands of high B concentration around regions of high A concentration.

This is only one of the reasons that ODEs fail. 4 Other important aspects

This is only one of the reasons that ODEs fail. 4 Other important aspects that ODEs fail to describe are: 4 The effect of delays (ODEs assume that the results of any interaction is immediate). 4 The limited capacity of space (ODES usually assume point like objects) 4 …. .