Selection, large deviations and metastability Kyoto () Dynamics - - PowerPoint PPT Presentation

Selection, large deviations and metastability

Kyoto

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• 1. Dynamics with selection

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A cell performs complex dynamics: DNA codes for the production of proteins, which themselves modify how the reading is done. A bit like a program and its RAM content.

DNA contains about the same amount of information as the TeXShop program for Mac

This dynamics admits more than one attractor: same DNA yields liver and eye cells... The dynamical state is inherited. On top of this process, there is the selection associated to the death and reproduction of individual cells

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Stern, Dror, Stolovicki, Brenner, and Braun

An arbitrary and dramatic rewiring of the genome of a yeast cell: the presence of glucose causes repression of histidine biosynthesis, an essential process Cells are brutally challenged in the presence of glucose, nothing in evolution prepared them for that!

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Stern, Dror, Stolovicki, Brenner, and Braun

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Stern, Dror, Stolovicki, Brenner, and Braun

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the system finds a transcriptional state with many changes two realizations of the experiment yield vastly different solutions the same dynamical system seems to have chosen a different attractor which is then inherited over many generations

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If this interpretation is confirmed, we are facing a dynamics in a complex landscape with the added element of selection but note that fitness does not drive the dynamics, it acts on its results the landscape is not the ‘fitness landscape’

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• 2. The relation between

a) Large Deviations, b) Metastability c) Dynamics with selection and phase transitions

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a pendulum immersed in a low-temperature bath

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a pendulum immersed in a low-temperature bath

θ

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Imposing the average angle, the trajectory shares its time between saddles 0o and 180o

180 θ Θ(τ) τ

phase-separation is a first order transition!

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R D[θ]P(trajectory) δ hR t

0 θ(t0) dt0 tθo

i = R dλ Z D[θ]P(trajectory) eλ

R t

0 θ(t0) dt0

| {z }

canonical

eλtθo canonical version, with λ conjugated to θ Z(λ) = R D[θ]P(trajectory) eλ

R t

0 θ(t0) dt0 () Dynamics with selection, large deviations and metastability 13 / 36

• λ is fixed to give the appropriate θ (Laplace transform variable)
• a system of walkers with cloning rate λθ(t)

dP dt =

d

dθ

• T d

dθ + sin(θ)

• P λθ P

yields the ‘canonical’ version of the large-deviation function

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• the relation is useful for efficient simulations
• but also to understand the large deviation

function

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Relation with selection

We wish to simulate an event with an unusually large value of A without having to wait for this to happen spontaneously but without forcing the situation artificially

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N independent simulations with probability c . A per unit time kill or clone x x ... continue ...

a way to count trajectories weighted with ecA

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Dynamical phase transitions

large deviations of the activity

JP Garrahan, RL Jack, V Lecomte, E Pitard, K van Duijvendijk, and Frederic van Wijland

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Competition between colonies

=(escape time) x

A B

A

B

λ A λ B=A in

=A in A

τ λA λB + 1/τ

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• A collection of metastable states
• each with its own emigration rate
• and its cloning/death rates dependent upon the observable

One way to understand the relation between metastability and large deviations

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Large deviations with metastability as first order transitions: space time view

A dynamics: e.g. Langevin: ˙

xi = fi(x) + ηi

= add all trajectories with weight: S[x] = 1

T

R dt { ˙ xi + fi(x)}2...

For small T, all trajectories that stay in a metastable state ˙ xi = fi = 0 contribute ‘almost’ the same

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in detail

x t x

A B

cost ~ escape rate cost ~ 0 cost ~ \ln(escape time) (small!)

ice-water at -0.001 oC

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Large deviations and first order

Large deviation function heλ

R dtA[x]i =

R dλP(A)eλA = trajectories with weight:

SA[x] = 1

T

R dt { ˙ xi + fi(x)}2... + λA(x)

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The observable A chooses the phase, for λ just larger than the escape rate

x t x

A B

cost ~ escape rate cost ~ 0 cost ~ \ln(escape time) (small!)

A

+ A in +A in A B

Another way to understand the relation between metastability and large deviations

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Activity, ‘glass’ transition Garrahan and Jack

inactive

EA > 0

qEA > 0

T T T

K d

• qEA = 0

s T

active (metastable) active (paramagnet) (spin glass)

q

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Champagne cup potential - spherical coordinates

O(N)

A Langevin process for the radius: ˙ r = d

dr {V (N 1)T ln r}

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Champagne cup potential - Phase diagram

critical

T

s

‘liquid’

metastable

T T

‘solid’

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• 3. A model

G Bunin, JK

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M individuals. Attractors with timescale τa and reproduction rate λa

max

P( ) τ Q( ) λ λ τ λ τ

max

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Without selection pressure the population reaches a finite (smallish) hτi As soon as the λi are turned one, the stationary state dissappears hτi ! 1, and λ ⇠ λmax

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Evolution of attractor lifetime hτi(t) ⇠ t if P(τ) ⇠ τ α a power law with α > 2 hτi(t) ⇠ t

1 2

if P(τ) ⇠ eaτ hτi(t) ⇠ t

1 3

if P(τ) ⇠ eaτ 2 Population divergence time fitness/mutation-rate (anti)correlation tdiv ⇠ t if P(τ) ⇠ τ α a power law with α > 2 tdiv ⇠ t2 if P(τ) ⇠ eaτ, tdiv ⇠ t3 if P(τ) ⇠ eaτ 2,

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Aging curves

10

1

10

2

10

−5

10

−4

10

−3

10

−2

10

−1

10 t−t* Cinner−prod(t−t*) () Dynamics with selection, large deviations and metastability 33 / 36

Fraction of population at t born before t⇤

20 40 60 80 100 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 t* Capprox(t=100 , t*) () Dynamics with selection, large deviations and metastability 34 / 36

How can we understand this anti-intuitive result?

max

λmax

aging stationary

1/τ

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Most of the population stays in states with untypically large stability Average fitness of the population hardly improves with time At large times, lineages present at the beginning manifest themselves!

We may understand this from the large-deviation point

• f view

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