The Structure, Function, and Evolution of Biological Systems

The Structure, Function, and Evolution of Biological Systems

The Structure, Function, and Evolution of Biological Systems

Instructor: Van Savage

Spring 2010 Quarter

4/27/2010

Instructor: Van Savage

Spring 2010 Quarter

4/27/2010

OutlineOutline

1. Finite size populations and genetic drift

2. Coalescence

3. Understanding directional and nondirectional forces

3. General Diffusion Equation

4. Biology

a. Bacterium’s use of 3 (physical constraint)

b. Population genetics--combining selection and drift in evolution (Conceptual analogy)

Derive equations in this context

1. Finite size populations and genetic drift

2. Coalescence

3. Understanding directional and nondirectional forces

3. General Diffusion Equation

4. Biology

a. Bacterium’s use of 3 (physical constraint)

b. Population genetics--combining selection and drift in evolution (Conceptual analogy)

Derive equations in this context

Genetic Drift--Nondirectional force

Genetic Drift--Nondirectional force

Drift--by random, transient, non-genetic events, individuals that would be highly reproductive in a “repeated” experiment, are lost to death.

Imagine a population of identical individuals that are being chosen for mating. Random chance that a given individual will be chosen.

Drift--by random, transient, non-genetic events, individuals that would be highly reproductive in a “repeated” experiment, are lost to death.

Imagine a population of identical individuals that are being chosen for mating. Random chance that a given individual will be chosen.

Genetic DriftGenetic DriftModel this by randomly sampling from entire population

(Wright-Fisher model). Population size, N, is constant, and individuals are randomly selected for mating

Each individual has 1/N chance of reproducing.We get a binomial tree that depends on frequency, p, and total

population size, N.

Model this by randomly sampling from entire population (Wright-Fisher model). Population size, N, is constant, and individuals are randomly selected for mating

Each individual has 1/N chance of reproducing.We get a binomial tree that depends on frequency, p, and total

population size, N.

Generation 0

Genetic DriftGenetic Drift

time

frequency, p

1

So, rate of spread of the width of distribution is~p(1-p)/2N

Coalescence: Look backwards in timeCoalescence: Look backwards in time

Quick resultsQuick results

1. P(fixation)=p

2. P(fixation new individual mutant)=1/2N

3. Total fixation rate of mutations=2Nμ(1/2N)=μ

4. Probability of coalescing t+1 generations

back is (1/2N)e-t/2N

1. P(fixation)=p

2. P(fixation new individual mutant)=1/2N

3. Total fixation rate of mutations=2Nμ(1/2N)=μ

4. Probability of coalescing t+1 generations

back is (1/2N)e-t/2N

Using last resultUsing last result

1. <T>=2N

2. StDev(T)=2N

1. <T>=2N

2. StDev(T)=2N

General diffusion equationand

combining selection and drift

General diffusion equationand

combining selection and drift

Directional Forces: Go with the flow

Directional Forces: Go with the flow

Directional ForcesDirectional Forces

1. Crowd running towards a celebrity or away from a fire.

2. Pushing or rolling any ball or object

3. A river flowing towards the sea or ocean

1. Crowd running towards a celebrity or away from a fire.

2. Pushing or rolling any ball or object

3. A river flowing towards the sea or ocean

Position

t0 t1 t2 t3 t4

Nondirectional forces: No flowNondirectional forces: No flow

Nondirectional forcesNondirectional forces

v

1. Lost in a crowded intersection2. Drop of dye in water3. Smoke4. Choosing each step after flipping a coin

Net Flow--Directional ForcesNet Flow--Directional Forces

f(t,i) is abundance of or probability of being in bin i at time, t.

v(i) is speed of flow out of bin i.

f(t,i) is abundance of or probability of being in bin i at time, t.

v(i) is speed of flow out of bin i.

i-1 i i+1

Net Flow=Flow In-Flow Out =Flow from left(i-1->i) - Flow to right(i->i+1)->f(t+1,i)-f(t,i)=v(i-1)f(i-1,t)-v(i)f(i,t)Continuum limit:->df/dt=-d(vf)/dx (i.e., distance=velocity*time)

x

f equilibrium

boundary(wall)

Net Flow--Nondirectional ForcesNet Flow--Nondirectional Forces

i-1 i i+1

Net Flow=Flow In-Flow Out =Flow from left (i-1->i)+Flow from right (i+1->i) -Flow to right (i->i-1)-Flow to left (i->i+1) ->f(t+1,i)-f(t,i)=D(i-1)f(i-1,t)+D(i+1)f(i,t)-2*D(i)f(i,t) D(i) is the diffusion rate

Continuum limit: ->df/dt=d2(Df)/dx2 (Second derivative)

Local process and affects width of distribution, not mean

x

fEquilibrium (D=constant)

Global signature of diffusionGlobal signature of diffusion Random walk x(t+1)=x(t)±1 ->x2(t+1)=x2(t)+2x(t)+1(1/2 of time) =x2(t)-2x(t)+1(1/2 of time)

<x2(t+1)>=(1/2)* <(x2(t)+2x(t)+1)> +(1/2)*<(x2(t)-2x(t)+1)> =<x2(t)>+1=<x2(t-1)>+2 Iterating this gives: <x2(t+1)>=Number of time steps~t

Random walk x(t+1)=x(t)±1 ->x2(t+1)=x2(t)+2x(t)+1(1/2 of time) =x2(t)-2x(t)+1(1/2 of time)

<x2(t+1)>=(1/2)* <(x2(t)+2x(t)+1)> +(1/2)*<(x2(t)-2x(t)+1)> =<x2(t)>+1=<x2(t-1)>+2 Iterating this gives: <x2(t+1)>=Number of time steps~t

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⇒ | x |= x 2 = t

Diffusion properties and simulation

Diffusion properties and simulation

http://web.mit.edu/course/3/3.091/www/diffusion/

Internal dynamics of diffusion are not immediately obvious from global behaviors, whereas it is for directional forces

Nondirectional force (diffusion) affects width of distribution, and directional forces affect the mean.

Diffusion process is NOT time reversible. Initial conditions are forgotten.

http://web.mit.edu/course/3/3.091/www/diffusion/

Internal dynamics of diffusion are not immediately obvious from global behaviors, whereas it is for directional forces

Nondirectional force (diffusion) affects width of distribution, and directional forces affect the mean.

Diffusion process is NOT time reversible. Initial conditions are forgotten.

2. Combined Effects2. Combined EffectsPerson trying to walk north (directional) through a

busy intersection (nondirectional)

Net Flow=Directional Flow+Nondirectional Flow

Person trying to walk north (directional) through a busy intersection (nondirectional)

Net Flow=Directional Flow+Nondirectional Flow

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∂f∂t

= −∂(vf )

∂x+∂ 2(Df )

∂x 2

Diffusion Equation(Also known as Kolmogorov forward equation)

Often, v and D are constant, so:Often, v and D are constant, so:

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∂f∂t

= −v∂f

∂x+ D

∂ 2 f

∂x 2

3. Physics--Brownian Motion3. Physics--Brownian Motion

Molecule in glass of water is analogous to our person walking through a crowd.

Since molecule is so small (mass is so little), gravity’s effect (directional force) is negligible. Hence,

Molecule in glass of water is analogous to our person walking through a crowd.

Since molecule is so small (mass is so little), gravity’s effect (directional force) is negligible. Hence,

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∂f∂t

= D∂ 2 f

∂x 2

(Heat Equation)

4. Biology—Magnetotactic Bacteria4. Biology—Magnetotactic Bacteria

Magnetotactic BacteriaMagnetotactic Bacteria

Weigh so little that gravity is negligible, andthey do not know which way is down.

Better conditions at bottom(oxygen pressure)

They have internalized enough magnetiteparticles so that earth’s magnetic field can justovercome nondirectional forces of Brownian Motion. Since magnetic fields go into earth, they can now sense down. So, they “solve” problemfrom previous slide! Bacteria in north and south are polarized differently.

Apply magnetic field to diffusing particles

Apply magnetic field to diffusing particles

This gives a directional force, and since magnetic force is much stronger than gravity, this is not negligible. Must return to full equation.

This gives a directional force, and since magnetic force is much stronger than gravity, this is not negligible. Must return to full equation.

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∂f∂t

= −∂(vf )

∂x+∂ 2(Df )

∂x 2

v depends on strength of magnetic field

Types of multidisciplinary influences

Types of multidisciplinary influences

Physical Process Conceptual and Mathematical Analogy

Magnetotactic bacteria Evolution and Population Genetics-> Combine natural selection and genetic drift

4. B. Combining Selection and Drift

4. B. Combining Selection and Drift

We can also understand process of evolution by means of diffusion equation. Requires different sort of extension to biology. It’s not just understanding how biological organisms use and are constrained by physics, but it’s using analogies to mathematical physics to understand biological problems.

We can also understand process of evolution by means of diffusion equation. Requires different sort of extension to biology. It’s not just understanding how biological organisms use and are constrained by physics, but it’s using analogies to mathematical physics to understand biological problems.

Selection--Directional ForceSelection--Directional ForceLet a population (wild type) suddenly have a few

individuals with a mutation that forms a new allele.If fitness (as measured by growth rate--number of

offspring per individual per generation that survive to next generation) of wild type is normalized to 1, and mutants have fitness 1+s

Let a population (wild type) suddenly have a few individuals with a mutation that forms a new allele.

If fitness (as measured by growth rate--number of offspring per individual per generation that survive to next generation) of wild type is normalized to 1, and mutants have fitness 1+s

Population Size

time

Wild type

Mutant

Position space, x, is replaced by frequency space, p, for frequency of mutants.Velocity of selection force is~ p(1-p)s

time, t

frequency of mutants, p

1

If population size is fixed (finite resources), only a matter of time, until mutant takes over (fixation).

Genetic DriftGenetic Drift

time

frequency, p

1

So, rate of spread of the width of distribution is~p(1-p)/N

Strong AnalogyStrong Analogy

1. Selection Gravity Pushed by marathon

2. Drift Brownian motion Crowd at intersection

3. Small organisms Small populations

(Brownian>>Gravity) (Drift>>Selection)

4. Large organisms Large populations

(Gravity>>Brownian) (Selection>>Drift)

1. Selection Gravity Pushed by marathon

2. Drift Brownian motion Crowd at intersection

3. Small organisms Small populations

(Brownian>>Gravity) (Drift>>Selection)

4. Large organisms Large populations

(Gravity>>Brownian) (Selection>>Drift)

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Equation for Population GeneticsEquation for Population Genetics

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∂P(p, t | p0)

dt= p(1− p)s

∂P(p, t | p0)

∂p0

+p(1− p)

2N

∂ 2P(p, t | p0)

∂p02

Questions we can answer using this equation.1. Is mutant population likely to go extinct or take over population (fixation)?2. How long does it take before extinction or fixation occurs?3. For a given N and s, how large does p0 need to be before mutants are likely to take over.

p0 is initial frequency of mutants in the population.

More proper derivationMore proper derivation

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Ψ(p, t + dt) = Ψ(p−ε, t)g(p−ε,ε,dt)dε∫

Taylor expand in p around epsilon to get Kolmogorov forward equations

probability densityof having frequencyp at time t+dt

Probability of movingfrom p-ε to p

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∂Ψ(p, t | p0)

dt= −

∂

∂p0

Ψ(p, t)M(p)[ ] +1

2

∂ 2

∂p02 Ψ(p, t)V (p)[ ]

More proper derivationMore proper derivation

Looking backward in time, as or coalescence gives Kolmogorov backward equation

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∂Ψ(p, t | p0)

dt= M(p)

∂Ψ(p, t)

∂p0

+V (p)

2

∂ 2Ψ(p, t)

∂p02

Sign of directional term flips because nowgoing backwards in times and is time reversible. Non-directional term does not flip sign becausenon-reversible.

Probability of FixationProbability of Fixation

Solve equation at and impose boundary condition for p0=0 and p0=1.

Solve equation at and impose boundary condition for p0=0 and p0=1.

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∂P∂t

= 0

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u(p0) =1− e−2Nsp0

1− e−2Ns

Probability of Fixation of mutants

Investigate some limitsInvestigate some limits

1. Large population, strong selection: e-2Nsp<<1 -> u(p0)~1 (guaranteed to fix)

2. Large population, really weak selection such that 2Nsp0<<1:e-2Nsp~1-2Nsp0-> u(p0)~2Nsp0

When one mutant, p0=1/N,and u(1/N)~2s (fixation probability increases linearly with s)

3. Under very weak selection (s->0):e-2Nsp~1-2Nsp0-> u(p0)~ p0,When one mutant, p0=1/N,and u(1/N)~1/N (same as for pure drift)

All limits check out.

5. Economics Black-Scholes model5. Economics Black-Scholes model

Price of stock is like position space (physics) or frequency space (population genetics).

Directional force--general increase in worth of the market, represented by interest rate.

Nondirectional force--random forces in market. Individual stocks or groups of stock will wander randomly in price. (major insight of this model!

Also, because it shows value of volatility and how to make money from it.)

Price of stock is like position space (physics) or frequency space (population genetics).

Directional force--general increase in worth of the market, represented by interest rate.

Nondirectional force--random forces in market. Individual stocks or groups of stock will wander randomly in price. (major insight of this model!

Also, because it shows value of volatility and how to make money from it.)

Assumptions of Black-ScholesAssumptions of Black-Scholes

1. Price follows Brownian motion

2. It is possible to short sell stock (options)

3. No arbitrage is possible (no asymmetry of which to take advantage)

4. Trading is continuous

5. No transactions costs or taxes

6. Stock’s price is continuous and can be arbitrarily small

7. Risk-free interest rate is constant

1. Price follows Brownian motion

2. It is possible to short sell stock (options)

3. No arbitrage is possible (no asymmetry of which to take advantage)

4. Trading is continuous

5. No transactions costs or taxes

6. Stock’s price is continuous and can be arbitrarily small

7. Risk-free interest rate is constant

Results from Black-ScholesResults from Black-Scholes

Provides method for calculating fair cost of an option.

Provides method for hedging “bets” and getting risk-free investment that allows one to make money according to the overall growth of the market.

Provides method for calculating fair cost of an option.

Provides method for hedging “bets” and getting risk-free investment that allows one to make money according to the overall growth of the market.

Black-Scholes PDEBlack-Scholes PDE

S-stock price

V-option cost

r-interest rate

-variance of random process

S-stock price

V-option cost

r-interest rate

-variance of random process€

∂V∂t

= −1

2σ 2S2 ∂

2V

∂S2− rS

∂V

∂S+ rV

Impact of this workImpact of this workOne equation, similar concepts, applications to multiple

fields with its own set of insights

1. Fourier developed the heat equation2. Brownian motion, Einstein’s greatest achievement?3. Applied in cosmology, particle physics, etc.4. Huge advance in population genetics, used to study

molecular motors, and lots of intracellular processes

5. 1997 Nobel prize in economics for Black-Scholes

One equation, similar concepts, applications to multiple fields with its own set of insights

1. Fourier developed the heat equation2. Brownian motion, Einstein’s greatest achievement?3. Applied in cosmology, particle physics, etc.4. Huge advance in population genetics, used to study

molecular motors, and lots of intracellular processes

5. 1997 Nobel prize in economics for Black-Scholes

ConclusionsConclusions1. Diffusion equations describe directional and nondirectional

forces. (Could also have forces on higher-order moments by extending this.)

2. Because of generality of 1, we can apply them to many different types of problems in many different fields. (Multidisciplinary)

3. Diffusion equations have already proved very useful in physics, biology, and economics.

4. Examples of two types of multidisciplinary science: a. Results from one field directly place constraints on or are

utilized by agents in the other field. b. By the correct choice of analogy between fields,

mathematical treatments and results can be used to draw new conclusions and insights within another field.

1. Diffusion equations describe directional and nondirectional forces. (Could also have forces on higher-order moments by extending this.)

2. Because of generality of 1, we can apply them to many different types of problems in many different fields. (Multidisciplinary)

3. Diffusion equations have already proved very useful in physics, biology, and economics.

4. Examples of two types of multidisciplinary science: a. Results from one field directly place constraints on or are

utilized by agents in the other field. b. By the correct choice of analogy between fields,

mathematical treatments and results can be used to draw new conclusions and insights within another field.