Selection, large deviations and metastability Kyoto () Dynamics with selection, large deviations and metastability 1 / 36
1. Dynamics with selection () Dynamics with selection, large deviations and metastability 2 / 36
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 () Dynamics with selection, large deviations and metastability 3 / 36
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! () Dynamics with selection, large deviations and metastability 4 / 36
Stern, Dror, Stolovicki, Brenner, and Braun () Dynamics with selection, large deviations and metastability 5 / 36
Stern, Dror, Stolovicki, Brenner, and Braun () Dynamics with selection, large deviations and metastability 6 / 36
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 () Dynamics with selection, large deviations and metastability 7 / 36
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’ () Dynamics with selection, large deviations and metastability 8 / 36
2. The relation between a) Large Deviations, b) Metastability c) Dynamics with selection and phase transitions () Dynamics with selection, large deviations and metastability 9 / 36
a pendulum immersed in a low-temperature bath () Dynamics with selection, large deviations and metastability 10 / 36
a pendulum immersed in a low-temperature bath θ () Dynamics with selection, large deviations and metastability 11 / 36
Imposing the average angle, the trajectory shares its time between saddles 0 o and 180 o Θ ( τ ) 180 θ 0 τ phase-separation is a first order transition! () Dynamics with selection, large deviations and metastability 12 / 36
hR t i R 0 θ ( t 0 ) dt 0 � t θ o D [ θ ] P ( trajectory ) δ Z R t R 0 θ ( t 0 ) dt 0 D [ θ ] P ( trajectory ) e λ e � λ t θ o = d λ | {z } canonical canonical version, with λ conjugated to θ R t R 0 θ ( t 0 ) dt 0 D [ θ ] P ( trajectory ) e λ Z ( λ ) = () 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 ) � d � � dP T d dt = � d θ + sin( θ ) P � λθ P d θ yields the ‘canonical’ version of the large-deviation function () Dynamics with selection, large deviations and metastability 14 / 36
• the relation is useful for efficient simulations • but also to understand the large deviation function () Dynamics with selection, large deviations and metastability 15 / 36
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 () Dynamics with selection, large deviations and metastability 16 / 36
N independent simulations x x with probability c . A per unit time kill or clone ... continue ... a way to count trajectories weighted with e cA () Dynamics with selection, large deviations and metastability 17 / 36
Dynamical phase transitions large deviations of the activity JP Garrahan, RL Jack, V Lecomte, E Pitard, K van Duijvendijk, and Frederic van Wijland () Dynamics with selection, large deviations and metastability 18 / 36
() Dynamics with selection, large deviations and metastability 19 / 36
Competition between colonies A B A τ =(escape time) x λ B =A in λ A B =A in A λ A � λ B + 1 / τ () Dynamics with selection, large deviations and metastability 20 / 36
• 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 () Dynamics with selection, large deviations and metastability 21 / 36
Large deviations with metastability as first order transitions: space time view A dynamics: e.g. Langevin: ˙ x i = � f i ( x ) + η i R = add all trajectories with weight: S [ x ] = � 1 x i + f i ( x ) } 2 ... dt { ˙ T For small T , all trajectories that stay in a metastable state x i = f i = 0 contribute ‘almost’ the same ˙ () Dynamics with selection, large deviations and metastability 22 / 36
in detail A B x x cost ~ escape rate cost ~ \ln(escape time) (small!) cost ~ 0 t ice-water at -0.001 o C () Dynamics with selection, large deviations and metastability 23 / 36
Large deviations and first order R Large deviation function h e λ R dtA [ x ] i = d λ P ( A ) e � λ A = trajectories with weight: R S A [ x ] = 1 x i + f i ( x ) } 2 ... + λ A ( x ) dt { ˙ T () Dynamics with selection, large deviations and metastability 24 / 36
The observable A chooses the phase, for λ just larger than the escape rate A B A x x cost ~ escape rate cost ~ \ln(escape time) (small!) + A in A +A in cost ~ 0 B t Another way to understand the relation between metastability and large deviations () Dynamics with selection, large deviations and metastability 25 / 36
Activity, ‘glass’ transition Garrahan and Jack T active (paramagnet) q EA = 0 T o inactive (spin glass) T d active q EA > 0 (metastable) T K q EA > 0 s () Dynamics with selection, large deviations and metastability 26 / 36
Champagne cup potential - spherical coordinates O(N) r = � d A Langevin process for the radius: ˙ dr { V � ( N � 1) T ln r } () Dynamics with selection, large deviations and metastability 27 / 36
Champagne cup potential - Phase diagram T ‘liquid’ T metastable ‘solid’ T critical s () Dynamics with selection, large deviations and metastability 28 / 36
3. A model G Bunin, JK () Dynamics with selection, large deviations and metastability 29 / 36
M individuals. Attractors with timescale τ a and reproduction rate λ a λ τ Q( ) P( ) τ τ λ λ max max () Dynamics with selection, large deviations and metastability 30 / 36
Without selection pressure the population reaches a finite (smallish) h τ i As soon as the λ i are turned one, the stationary state dissappears h τ i ! 1 , and λ ⇠ λ max () Dynamics with selection, large deviations and metastability 31 / 36
Evolution of attractor lifetime if P ( τ ) ⇠ τ � α a power law with α > 2 h τ i ( t ) ⇠ t 1 if P ( τ ) ⇠ e � a τ h τ i ( t ) ⇠ t 2 1 if P ( τ ) ⇠ e � a τ 2 h τ i ( t ) ⇠ t 3 Population divergence time fitness/mutation-rate (anti)correlation if P ( τ ) ⇠ τ � α a power law with α > 2 t div ⇠ t t div ⇠ t 2 if P ( τ ) ⇠ e � a τ , t div ⇠ t 3 if P ( τ ) ⇠ e � a τ 2 , () Dynamics with selection, large deviations and metastability 32 / 36
Aging curves 0 10 − 1 10 C inner − prod (t − t*) − 2 10 − 3 10 − 4 10 − 5 10 1 2 10 10 t − t* () Dynamics with selection, large deviations and metastability 33 / 36
Fraction of population at t born before t ⇤ 1 0.95 0.9 C approx (t=100 , t*) 0.85 0.8 0.75 0.7 0.65 0 20 40 60 80 100 t* () Dynamics with selection, large deviations and metastability 34 / 36
How can we understand this anti-intuitive result? 1/ τ max stationary aging λ max () Dynamics with selection, large deviations and metastability 35 / 36
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 of view () Dynamics with selection, large deviations and metastability 36 / 36
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