Selection, large deviations and metastability€¦ · Selection, large deviations and metastability Kyoto Dynamics with selection, large deviations and metastability 1 / 36. 1. Dynamics
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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 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
() 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 histidinebiosynthesis, an essential process
Cells are brutally challenged in the presence of glucose, nothingin 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 differentsolutions
the same dynamical system seems to have chosen adifferent 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 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’
() 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
!
() Dynamics with selection, large deviations and metastability 11 / 36
Imposing the average angle, the trajectory shares its timebetween saddles 0o and 180o
180
!
"(#)
#
0
phase-separation is a first order transition!
() Dynamics with selection, large deviations and metastability 12 / 36
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
() Dynamics with selection, large deviations and metastability 13 / 36
• � 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
() Dynamics with selection, large deviations and metastability 14 / 36
• the relation is useful for efficient simulations
• but also to understand the large deviationfunction
() 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
with probability c . A per unit time kill or clone
x x
... continue ...
a way to count trajectories weighted with e
cA
() Dynamics with selection, large deviations and metastability 17 / 36
Dynamical phase transitionslarge 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
=(escape time)
x
A
BA
B!A !B=A in =A in A
"
�
A
� �
B
+ 1/⌧
() 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 betweenmetastability and large deviations
() Dynamics with selection, large deviations and metastability 21 / 36
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
() Dynamics with selection, large deviations and metastability 23 / 36
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)
() Dynamics with selection, large deviations and metastability 24 / 36
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
() Dynamics with selection, large deviations and metastability 25 / 36
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
() Dynamics with selection, large deviations and metastability 26 / 36
Champagne cup potential - spherical coordinates
O(N)
A Langevin process for the radius: r = � d
dr
{V � (N � 1)T ln r}
() Dynamics with selection, large deviations and metastability 27 / 36
Champagne cup potential - Phase diagram
critical
T
s
‘liquid’
metastableT
T
‘solid’
() 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 ⌧
a
and reproductionrate �
a
max
P( )!Q( )"
" !" !max
() Dynamics with selection, large deviations and metastability 30 / 36
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
() Dynamics with selection, large deviations and metastability 31 / 36
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,
() Dynamics with selection, large deviations and metastability 32 / 36
Aging curves
101 10210−5
10−4
10−3
10−2
10−1
100
t−t*
Cinner−prod(t−t*)
() Dynamics with selection, large deviations and metastability 33 / 36
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
*)
() Dynamics with selection, large deviations and metastability 34 / 36
How can we understand this anti-intuitive result?
max
!max
aging
stationary
1/"
() Dynamics with selection, large deviations and metastability 35 / 36
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
() Dynamics with selection, large deviations and metastability 36 / 36
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