Selection, large deviations and metastability€¦ · Selection, large deviations and metastability...

Selection, large deviations and metastability

Kyoto

() Dynamics with selection, large deviations and metastability 1 / 36

1. Dynamics with selection


A cell performs complex dynamics: DNA codes for theproduction of proteins, which themselves modify how thereading 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 DNAyields liver and eye cells...

The dynamical state is inherited.

On top of this process, there is the selection associated tothe death and reproduction of individual cells


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 histidinebiosynthesis, an essential process

Cells are brutally challenged in the presence of glucose, nothingin evolution prepared them for that!






the system finds a transcriptional state with many changes

two realizations of the experiment yield vastly differentsolutions

the same dynamical system seems to have chosen adifferent attractor which is then inherited over many generations


If this interpretation is confirmed, we are facing a dynamics in acomplex landscape

with the added element of selection

but note that fitness does not drive the dynamics, it acts on itsresults

the landscape is not the ‘fitness landscape’


2. The relation between

a) Large Deviations,

b) Metastability

c) Dynamics with selection and phase transitions


a pendulum immersed in a low-temperature bath


a pendulum immersed in a low-temperature bath

!


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

180

!

"(#)

#

0

phase-separation is a first order transition!


RD[✓]P (trajectory) �

hRt

0 ✓(t0) dt0 � t✓

o

i

=Rd�

ZD[✓]P (trajectory) e

�

Rt

0 ✓(t0) dt0

| {z }canonical

e

��t✓

o

canonical version, with � conjugated to ✓

Z(�) =RD[✓]P (trajectory) e

�

Rt

0 ✓(t0) dt0


• � is fixed to give the appropriate ✓ (Laplace transform variable)

• a system of walkers with cloning rate �✓(t)

dP

dt

= ��

d

d✓

�T

d

d✓

+ sin(✓)�

P � �✓ P

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


• the relation is useful for efficient simulations

• but also to understand the large deviationfunction


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


N independent simulations

with probability c . A per unit time kill or clone

x x

... continue ...

a way to count trajectories weighted with e

cA


Dynamical phase transitionslarge deviations of the activity

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

Frederic van Wijland



Competition between colonies

=(escape time)

x

A

BA

B!A !B=A in =A in A

"

�

A

� �

B

+ 1/⌧


• 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 betweenmetastability and large deviations


Large deviations with metastability as first ordertransitions: space time view

A dynamics: e.g. Langevin: x

i

= �f

i

(x) + ⌘

i

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

Rdt {x

i

+ f

i

(x)}2...

For small T , all trajectories that stay in a metastable statex

i

= f

i

= 0 contribute ‘almost’ the same() Dynamics with selection, large deviations and metastability 22 / 36

in detail

x

t

x

A

B

cost ~ escape rate

cost ~ 0

cost ~ \ln(escape time)

(small!)

ice-water at -0.001 o

C


Large deviations and first order

Large deviation function he�RdtA[x]i =

Rd�P (A)e��A

= trajectories with weight:

S

A

[x] = 1T

Rdt {x

i

+ f

i

(x)}2...+ �A(x)


The observable A chooses the phase, for � just larger than theescape rate

x

t

x

A

B

cost ~ escape rate

cost ~ 0

cost ~ \ln(escape time)

(small!)

A

+ A in

+A in

AB

Another way to understand the relation betweenmetastability and large deviations


Activity, ‘glass’ transition Garrahan and Jack

inactive

EA > 0

qEA > 0

T

T

TK

d

oqEA = 0

s

T

active(metastable)

active(paramagnet)

(spin glass)

q


Champagne cup potential - spherical coordinates

O(N)

A Langevin process for the radius: r = � d

dr

{V � (N � 1)T ln r}


Champagne cup potential - Phase diagram

critical

T

s

‘liquid’

metastableT

T

‘solid’


3. A model

G Bunin, JK


M individuals. Attractors with timescale ⌧

a

and reproductionrate �

a

max

P( )!Q( )"

" !" !max


Without selection pressure the population reaches a finite(smallish) h⌧i

As soon as the �

i

are turned one, the stationary statedissappears

h⌧i ! 1, and � ⇠ �

max


Evolution of attractor lifetime

h⌧i(t) ⇠ t if P (⌧) ⇠ ⌧

�↵

a power law with ↵ > 2

h⌧i(t) ⇠ t

12

if P (⌧) ⇠ e

�a⌧

h⌧i(t) ⇠ t

13

if P (⌧) ⇠ e

�a⌧

2

Population divergence timefitness/mutation-rate (anti)correlation

t

div

⇠ t if P (⌧) ⇠ ⌧

�↵

a power law with ↵ > 2

t

div

⇠ t

2if P (⌧) ⇠ e

�a⌧

,

t

div

⇠ t

3if P (⌧) ⇠ e

�a⌧

2,


Aging curves

101 10210−5

10−4

10−3

10−2

10−1

100

t−t*

Cinner−prod(t−t*)


Fraction of population at t born before t

⇤

0 20 40 60 80 100

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

t*

Cap

prox

(t=10

0 , t

*)


How can we understand this anti-intuitive result?

max

!max

aging

stationary

1/"


Most of the population stays in states with untypically largestability

Average fitness of the population hardly improves with time

At large times, lineages present at the beginning manifestthemselves!

We may understand this from the large-deviation pointof view


Date post:	19-Aug-2020
Category:	Documents
Upload:	others
View:	13 times
Download:	0 times

Selection, large deviations and metastability€¦ · Selection, large deviations and metastability...

Documents