Seminar in mathematical BiologyTheoretical issues in modeling

Yoram Louzoun

Nadav Shnerb

Eldad Bettelheim

Sorin Solomon

Overview Dynamical systems are traditionally modeled using

ODEs. However most of the assumptions under which

ODEs are correct are not valid in biological systems.

We analyze some prototypic systems and present a new method to analyze the behavior of stochastic spatially extended systems.

Using our results, we show that well known concepts, such as Malthusian growth and ecological niche are the result of over simplifications.

Fisher Equation

The simplest system contains a single agent and three processes: Birth, death and competition.

Such system is represented by a Fisher Equation.

is the birth rate- the death rate, is the competition rate

),(),(),( 22 txbDtxbtxbt

b

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

first solution is stable for (0 ) and the second solution is stable for ( >0).

If ( >0) the stable solution will invade the unstable solution forming a Fisher front with a width advancing with a velocity

Small fluctuation from the stable steady state of wavelength k decay at as exp[- (||+k2)t].

Fisher R.A 1937

/Dw

.2 Dv

Catalytic noise

In many biological and especially in social systems the creation of a new substance is induced by a catalyst.

The catalyst can have a very low density. This low density induces fluctuation in its local density.

The fluctuations in the catalyst density affects the reaction rates of the reaction they catalyze.

Full system

In order to asses the effect of such noise we model a spatially extended systems with a catalyst and a simple reaction.

We model an agent (B) proliferating in the presence of a catalyst (A) with a probability , and dying with a rate .

Both the catalyst and the agent are diffusing We ignore at this stage the non linear term.

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

A

B+

A

B B

B

Naive PDE The PDE describing this system is:

If we let the A diffuse for a long time, before B reactions starts the A distribution will be:

A(x)=A0

ADt

A

BxmBDBDBABt

B

A

BB

2

22 )(

Malthus

The solution to these equation is an extrapolation from Malthus:

c is a positive constant, and is 0 for the mode representing uniform initial conditions of B(x).

c

tcuAc exxB )()()(

Naïve PDE II

The fastest growing mode has an exponent of: -

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

We simulate this system, with a lattice constant l containing As and Bs.

Each point in the lattice can contain an infinite number of A and Bs.

Interaction are only within the same lattice point. The interaction rate is /ld

The diffusion probability from a lattice point to its neighbor is: D/l2

We simulate a case when the death rate is much higher than the birth rate: (A /ld -) < <0 .

Simulation Results

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

In order to understand this strange result, lets simplify our system. We will make it one dimensional with 18 cells.

Assume an A concentration of A=0.2, /l =1 and =0.5.

Even Simpler Example II

At t=0 The Bs have a random distribution . At t=1 Half the B disappeared and 20% of

new Bs were created. At t=2 the same process takes place…..

Even Simpler Example III

But this is not the social reality, the As are discrete entities.

At places where there are no As all the Bs will disappear.

At places where at least one A exists the Bs will prosper.

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

growth and a linear cut-off In an inhomogeneous situation, regions where the local

birth rate is lower than the death rate, the total population will simply shrink to zero.

In regions where the local birth rate is higher than the death rate the total population will increase to infinity.

After a finite time, only regions with a high B population will influence the average B population.

Single A Approximation

If only a single A exists, then the average A population is close to zero.

If the A is fixed in space then the B at the location of the A obeys:

tldDlBLn

BldDBBldt

dB

Bd

Bd

)/2/()(

/2/

2

2

But, What About Diffusion???

After a some time the A will diffuse, and all this story will not work any more.

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.

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

The size of the B island is growing linearly.

The As on the other hand diffuses, so that they move with a rate proportional to the square root of t.

tlD

ldDlRR

dB

Bd

)/ln(

)/2/()0(

2

2

2/)( ltDtS A

Island Shape The neighboring cells have a constant input

from the cell containing the A and the same diffusion and death rate:

)(/2)(

)0(/)(

2

)/2/(2 2

rBldDrB

eBlDdt

rdB

B

tldDlB

Bd

)/ln())(( 2

d

BlDr

rBLn

A jumps

When A jumps to a neighboring cell, the Ln of the B population at the A position decreases by:

The population at the center of the island is thus:

)/ln( 2 dBlD

tlDldD

ldDlBLnd

BA

Bd

))/ln(/2

/2/()(22

2

Emerging Island

Cooperative effects

We simulate a system with =0.5, =1, and <A>=0.5

( <A>- )=-0.75 Although in lattice points containing a single A

the B population decreases. In points containing 3 or more As the B population increases.

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

The A concentration has a Poisson distribution.

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.

!1

mA

eVm

A

Low A diffusion

For any value of , and DB, there is a value m that will obey:

The B population will grow precisely in these points.

In other words, in the low A diffusion regime the B population grows following the maximum of A and not its average.

0/2/ 2 ldDlm Bd

Low A diffusion

Master Equation

The situation of the system can be described using a master equation for the probability to have A1 As and B1 Bs in the first cell… P(A1B1..ANBN).

The master equation for a single point on the lattice is:

termDiffusionPmmP

PmnmnPPH

PHdt

dP

mnnm

mnnm

nm

])1([

])1([)(

)(

)1(

)1(

Translation to quantum formalism

In order to solve this equation,we replace the classical probability function by a quantum wave function:

>=Pnm/n!/m!(a†)n(b †)m|0>. We define creation and destruction of A and Bs

using the quantum creation and annihilation operators: a, a†,b, b† .

Adding a new A particle is simply: a†| >, and destroying a particle is: a| >.

Creation and annihilation.

These operators follows the following simple rules:

a†|n,m>= |n+1,m>; a|n,m>= n|n-1,m> b†|n,m>= |n,m+1>; b|n,m>= m|n,m-1> [a,b]=0 ; [a, a†]= a a†- a† a =1 Counting particles in a given location is: a†a|n,m>= n a†|n-1,m>=n|n,m>

Hamiltonian

We replace the Hamiltonian with a quantum Hamiltonian, and obtain a non imaginary schroedinger equation.

’=H

nneii

b

nneii

a

iiiiiidiii

bbbl

Daaa

l

D

bbbaabbaal

bbbH

)()(

][][

†2

†2

†i

†i

††i

††

Mean field

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

bbl

naal

bbaa

a

iiii

22

††

1 ,

1

1 ,1

RG -2D

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

transition, and in some of phase space the ODEs are precise.

Local Competition

The Bs may compete over a local resource (food, space ,light…). This local competition is limited to Bs living on the same lattice site.

The local competition will mot change the overall dynamics, but it will limit the size and total population of each B island.

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

Global Competition

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.

For example cells competing for a resource in the blood, animals competing over water, plankton competing over oxygen.

Large scale competition is described as an interaction with the total population over some scale The competition reaction is :

B+<B>-><B>

The world company

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.

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.

The life-span of these islands is much larger than the life-span of a single B.

These emerging islands will lead to the creation of intermittent fluctuations in the total B population.

Predator prey systems

We have shown that the classical PDE treatment of a ver simple autocatalytic systems is wrong.

One of the reasons for the large difference is the high correlation between a and b fluctuations.

We will show that PDE fail in system with anti-correlation terms

Predator pray systems

Lets denote a pray by a, and a predator by b. The pray population grows, unless destroyed

by the predator. The predator population is growing when it

is “eating” the pray. The predator population is limited by death

(linear) and competition (non-linear)

Ecological niches

The equations describing the system are:

These equations have 2 fix points: (0,0) and ([/+]/,/) , but only the non zero fix

point is stable. This is the origin of the ecological niche

concept.

aDbbabb

aDabaa

a

a

22

2

Infection Dynamics-ODE

Infection Dynamics-ODE

Infection

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.

As in the previous cases, the ODEs fail to take into account some elemental biological features which makes their results obsolete.

Missing elements. 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.

We are explicitly simulating an immune system, but these elements are present in every P-P system.

Simulations.

Spatial distribution.

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.

One can ask what happens if the pathogen is presented in a single point of space.

Infection Dynamics Simulations.

Random Spatial structure

The immune system is dwelling in an Euclidean space.

A more realistic simulation should contain random neighbors.

Global destruction.

An even more interesting dynamics can take place if the predator has a preying range much larger than the prey diffusion radius.

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.

P-P delay

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

Summary ODEs fail completely in describing autocatalytic systems. The total population of an agent with a lower proliferation

rate than death will increase, in contradiction with the homogenous description.

The chance of survival are much more important in 2D than in 3D.

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.

Other important aspects that ODEs fail to describe are:

The effect of delays (ODEs assume that the results of any interaction is immediate).

The limited capacity of space (ODES usually assume point like objects)

…..