Geometry Genetics and Evolution
With Paul Francois
and V. Hakim (ENS Paris), F. Corson, A. Warmflash (RU)
and A. Brivanlou, A. Vonica (RU) work on Xenopus
•Can evolution be made predictive for genetic networks (in some small ways)
•Can dynamical systems illuminate the function of gene networks
Is evolution purely contingent ?
Gedankenexperiment :Gould, Wonderful life, 1989
Rewind evolutionary tape
Play itback
Underground discussion of convergent evolution
Odd exceptions or the rule??
Developmental Gene Regulatory Networks
Our approach : “inverse problem” or ~genetic screen
postulate function(fitness) ->networks ->experiments
experiments->networks->functions
Boulouri &DavidsonVirgin of Vladimir
Questions
Define fitness: reproductive fitness too removed from properties of one network.
Thus need quality measure for performance of network.
Mutations: rates obscure & impossible to go from genomic mutations to changes in network behavior
Resolution: Fitness: case by case discussion, use multiple functions and show equivalence
Mutations: change question ‘what can be found by gradient search’, look for good local minima, ~ 19th C Darwinism
Consider example anterio-posterior patterning in animals in detail and then other cases in summary.
Digression: Evolution of the eye
DE Nilsson, S. Pelger, Proc Roy Sci B, 256 1471 1994“A Pessimistic estimate of the time required for an eye to evolve”
photosurface --> invagination --> pinhole camera --> lens
fitness = acuitymutation: ~0.5% heritable variation in any quant. trait in population, ∆selection ~ 0.01/generation (--> 0.005% variation/generation)
Time < 106 generations
Limits to theory, eg bistability: (Francois&Hakim PNAS 2004)
A
B(A1,B1)
(A2,B2)
Bistability: In concentration plane of two proteins, any starting point tends to (A1 B1) or (A2 B2)
Fitness function ?1. What is ideal behavior (max fitness)2. How to measure proximity to ideal3. Do (1,2) affect the result?
One evolved network:
A represses gene b, A made cst rate, A+B dimers disappear --> bistability
One real network (lac system)
(Logically equivalent to A--| B, a + b --> inert, A constitutive)
Fitness for embryonic patterning (1)
ci (x)
x
Conc
entr
atio
n
Selector gene hypothesis:Define compartments/segments,tracks cell lineage, cell autonomous
‘Morphology’ -> network that positions the selector genes
Fitness for embryonic patterning (1Dim)
Require for fitness:
1. Assign a number to any collection of selector genes Ci(x)
2. Max. diversity... many selector genes expressed in embryo
3. Min. diversity for given x... (unique fate)
Fitness as mutual entropy:
P(i | x) = Ci(x) / ∑i Ci(x) (only relative concentrations matter)P(x) = 1/L (uniform probability on cells)
Fitness favors:1. ‘Max. diversity’ -> Max entropy, S1, of P(i): 2. ‘Min. diversity given x’ -> Min entropy, S2, P(i | x):
How to combine 2 terms?3. Assume gene duplication neutral -> fitness = -S1 + S2 = min. mutual entropy (i, x). Ci <=> x
For N selector genes fitness ≥ -log(N). (Best fitness is most negative)
Mutual entropy fitness
F=-log 1.5
F=-log 2
F=-log 3
F=-log 2.3
Methods
Interactions either activate & add, or repress & multiply
eg A auto-activates, repressed by R1 R2 (dropping csts), G(time)
G A
R1R2
A = max(G(t),An1
1 + An1)
11 + Rn2
1
11 + Rn3
2
!A
Evolve gene networks via mutation-selection of both network topology and parameters
Embryo a line of cells
Network functions identically in all cells, which differ only in exposure to morphogen (external protein whose spacial profile determines fate), no direct cell-cell communication
= max(activators) *∏(repressors) - degradation
Networks for static ‘morphogen’
Morp On->Off Fitness(generation)
- ln(7)
Morpho
Networkgene
Selector
activaterepress
Final pattern, selectori(x)
Final network
Time (development) dynamics in evolved static morphogen network
Morpho
Networkgene
Selector
Properties of Networks static gradient
activaterepress
Networks ‘cell autonomous’ (no communication between cells)-> morphogen defines cell position.
Morphogen disappears -> multi-stability -> sharp boundaries & only need repression between ~adjacent domains
Multi-stability -> order of gene expression matters & numbers determine final state.
Morphogen sets anterior boundaries, repression sets posterior boundaries -> statistical char. of evolved networks.Morpho
Networkgene
Selector
Anterior-Posterior patterning
Hox genes conserved in bilaterians
Define coarse AP coordinates
Cellular “Zip code” controls master regulatory genes
Biochem of regulation very complex, but simple phenomenology
Phenomenology of Hox expression
1.Spacial colinearity: 3’ to 5’ genome order follows A to P expr.2.Temporal colinearity: (vertebrates) temporal order follows A to P3.Posterior prevalence rule: most posterior Hox gene imposes fate on all anterior genes
Hox mutation haltere->wing
Hox3mutateHox1
wingwing
wing = (1 AND NOT(2,3..))haltere = (2 AND NOT(3,..))
Hox3Hox2Hox1
winghaltere
Hox expression A to P
Xenopus development (2)
1.2mm egg, 7hrs stage 9 4000+ cells; 17hrs stage 15; 40hrs stage 32 @23C
Development of Xenopus
1.2mm egg
5 hrs fertilization to Movie04000+ cells
17hrs @23C Movie
Anterior
Posterior
Dorsal view
Gene expression delimits territories
Steiner AB etal Dev. 2006. Stage 10.25 images
Model ‘patterning during growth’ as sliding morphogen that marks boundary between growth zone and patterned tissue.
Patterning a field of cells : AP growth
‘organizer’ is point where converging equator -> extending AP
morphogen step ~ organizer
“Anterior” “Posterior”
Hox expression marked as colors.Temporal sequence of expression on equator->spacial domains AP
Wacher 2004
Sliding Morphogen
...
Bias initial state with two selector genes 1,2, and a gene 3 to lag production of 1.
Then evolve readily multiple patterning networks
Sliding morphogen (2)
Time development of final evolved network
only trianglesenter fitness
Morphogen
Mouse vertebrae reflect Hox territories
to five fused caudal vertebrae,whichmake up the specialized coccyx comparedto approximately 30 caudal vertebrae that make up the tail in mice.
2.2. Hox expression
The earliest indication that vertebrate Hox genes might play a role in verte-brate axial patterningwas obtained by in situ hybridization analyses.Hox genesare expressed from 30 to 50 in the clusters, with the earliest genes expressed inthe posterior primitive streak at late streak stages, and more 50 genes expressedat progressively later stages (Dressler and Gruss, 1989; Duboule and Dolle,1989; Gaunt, 1991; Gaunt and Strachan, 1996; Gaunt et al., 1986, 1990;Graham et al., 1989; Izpisua-Belmonte et al., 1991). This temporal control ofHox expression onset, coupled with growth and elongation of the embryo,results in spatially graded anterior boundaries of expression where 30 genes(Hox1 andHox2) display anterior expression limits in the hindbrain region ofthe embryo and increasingly 50 genes demonstrate increasingly posterior limits
C1 C2 C3 C4 C5
Cd7 Cd6 Cd5 Cd4 Cd3 Cd2 Cd1 S4 S3 S2
C6 C7 T1 T2 T3 T4 T5 T6
T7
T8
T9
T10
T11
T12
T13
L1
L2
L3
L4
L5
L6
S1
Figure 9.2 A lateral view of an E18.5 mouse skeleton stained with Alcian blue andalizarin red; anterior to the left, posterior to the right. Circumferentially around theedges of the panel, the individual vertebral elements are pictures, beginning with thefirst cervical vertebra, the atlas, in the top, left side with the elements in order,clockwise. C, cervical, T, thoracic, L, lumbar, S, sacral and Cd, caudal; numbersreflect their position in the skeleton. (Only 7 of the approximately 30 caudal vertebraeare shown).
Hox Genes and Vertebrate Axial Pattern 261
Temporal colinearity
Temporal colinearity: Hox(time) fixed posterior cell --> Anterior-Posterior progression
Anterior Homeotic Mutation (2)
Properties of Networks with Sliding Gradient
Hox phenomenology: temporal colinearity, anterior homeotic mutation
Recall static morphogen: anterior boundaries positioned from morphogen. Analogue for sliding gradient?
Position == time exposed to morphogen:‘Timer’ gene 3 converts time in morphogen to morphogen level + cell autonomy. Static Morph <-> Sliding Morph.•Good for growth control, change all rates get same pattern (Deschamps etal timer ~ caudal)•Evolution of long from short germ band insects
Genes(time) 1 cell
Other systems evolved:
• Clocks and bistable systems: (Francois & Hakim PNAS 2004)
• Somitogenesis (eg vertebrae): (Francois, Hakim, ES Mol.Sys.Bio. 2007)
• Adaptation to temporal signal (Francois & ES, Phys.Bio. 2008)
• Mutual entropy fitness (mutual inhibition no Morpho)
• Segment on spatial gradients (spatial adaptation)
• Noisy signal transduction...
Generalities?
Somitogenesis: another example of patterning during growth
• Patterning during growth, different fitness than A P Hox, but process happens parallel
•Stereotypic evolutionary path to network whose dynamics has immediate topological interpretation.
Fitness fn for Segmentation (1)
In most animal species, the body axis progressively forms by posterior elongation
Heart shape
Segmentedthorax
Fully segmented
TGZ
0 somites
4 somites
17 somites
Rostral
Caudal
PSM
Somites
SomitesPSM
TailBud
(Movie : Pourquié’s lab)Most arthropods
Chick
Segmentation in zebrafish (DllC stained)
J Lewis, Curr Bio 2003
FGF8 morphogenmoving with tailDubrulle Science
2004
Fitness function for segmentation (2)
Protein E defines segments. Fitness (F) = number of steps between E hi, E lo
E
position
F=3
E
position
F=2E
position
F=1
E
position
F=0(noise)
E
position
F=0(ramp)
Final Somitogenesis network (clock and wavefront)
Morphogen G, bistable E, delayed feedback oscillator R
Path of segmentation
Create a bistable system by + feedback, G hi, E->1, G lo E->0, necessary to ‘remember’ G, F=1
Create repressor for G > thresh E, -> one stripe, F=2
Negative feedback (with transl delay) of R creates oscillator and many stripes in one ‘mutation’
• Rates do not matter to path if transitions sequential, ‘fitness funnel’
• Several solutions but similar phenotype
• Phase space 4D: G(t), E, Oscillator.Note competing activities on E, push over/under bistability threshold
Only model of complete process. Fit genes to parameters
Property of these evolutionary simulations
g1
g2
g3
g4
Temporal Adaptation (PF, EDS Phys. Bio. 5 2008)
How to show that the fitness function does not matter??
Fitness function --> fitness space
(See also Behar .. Bio. Phys. J. 93, 806-21 2007 & PA Iglesias)
Adaptation (defn 1)
Output returns to cst for any input
Adaptation (defn 2: Metric)
Time
Input
I1
I2
ΔOss
ΔOmaxOutput
Definition: cst I -> cst O; change-Osteady-state ~0; change-Omax largeOptions: ΔOmax ~ I2/I1 or I2 - I1 and for what range of INB: All adapting networks require some constraint on parameters
Adaptive networks from Evolution (1)*
Ligand (input, I) + Receptor(R) <--> Active-Receptor(O)
dR/dt = 1 - (I R - O) + (0 R) (= prod. + kinetics + no-decay)dO/dt = (I R - O) - r O (= kinetics + decay)
Make receptor at rate 1, stable until binds ligandActive receptor is output, and unstable.Fixed point, I cst: 1 - r O == 0
Examples: GPCR-Kinase (platform for arrestin), fast, specific for active receptors, production of receptors via endocytosis recycling
*Conventions: uninteresting csts ->1, terms that must vanish indicted as 0*
Adaptive network (2)
I is enzyme that interconverts O and P viaMichaelis-Menton
dO/dt = 1 - MM(I,O,P) - r OdP/dt = MM(I,O,P) - (0 P)
Buffer level of O: 1 == r O, independent of I
These components (and others) can be combined in series and parallel to incorporate other properties, dynamic range….eg
O P
I Michaelisrate (MM)
r O
a
Rectify Output (report I2 < I1)
TF TF*
I
O
(transcription
Add new output down stream of previousand drive via transcription
Adaptive network (3)
Impose: input a T.F.Add a PPI between O & R
Linearized limit:dO/dt ~ I(t) - (O/(cst + O))*cst*I(t-lag) - (0*O)
Works with MM(I) but evol. favored if gene dupl. since inputs to O,R similar
Fitness fn --> Pareto optimality
TimeInput
I1
I2
ΔOss
ΔOmaxOutput
Desiderata: cst I -> cst O; change-Omax large; change-Osteady-state ~0, (max proteins small, time response rapid…)Tradeoffs among desiderata, we are ignorant
Pareto optimality: minimize two fns F1, F2
can not distinguish boundary
without tradeoffs F1 vs F2
parameters p1, p2, p3…
F1
F2
F1
F2
Pareto optimality and adaptation
Tradeoff between large ΔOmax and small ΔOss are species specific and not knowable.
BUT all previous networks, will optimize parameters via gradient descent.
ΔOss
ΔOmax
good
net
work
s adaptiveflows
Characteristics of evolved models
• Close to dynamical system picture, evolve topology of flow, not genes -> visualize minimal parameter description (-> genes to be fit)
• Networks work by sloppy confluence of opposing activities; with tuned rates; no time scale separation ≠ 19th C applied math. BUT simple in that parameters follow by gradient search.
• Volume in parameter space (of complete network) poor criterion; assumes network, ignores evolution, ~mutational load ∆fitness<< selection
• Evolved models not obvious, like screen
• Relevance to experiment, hi level (static <-> dynam morpho), lo level (fit parameters)
• No buzz words, robust, modular, evolvable...
• Evolution via incremental increases in fitness (positive selection, or grad. search) mitigates against dependence on mutation parameters.
• 19th C Darwinism -> grad search, Useful engineering for specific sys.
End
Anterior repression does not generalize
Atypical result of gene duplication:
Anterior boundary set by repression3--| 2
Misc figs
Minimal parameter description?
Normally: gene --> network --> parameters --> model But redundancies (among genes) and multiple time scales make reductionism hard.
Evolved model closer to minimal parameter set & suggestive of geometric description Genes couple to parameters in model
g1
g2
g3
g4
E.g. somitogenesis (vertebrae) :• Sliding morphogen gradient G• Bistable system E• Repressor R with delated neg. feedback to oscill.
--> 4 dimensional phase space
Evolved networks ≠ applied mathematics
Evolved networks often work by sloppy confluence of opposing activities eg activators and repressors, and opposing rates.
Dynamics essential & no separation of time scales.
BUT solutions found by gradient descent, in that sense simple
In-silico evolution like generic screen: ask interesting question (fitness) get interesting answer; guards against ex post facto reasoning, enumerates all ways of accomplishing task.
Volume in parameter space a poor tool for model selection:• ~ mutational load ~ mutation rates << O(1) vs. fitness changes• presumes ~final network, ignores evolution
Summary:
Can the ‘theory’ of evolution be made predictive in some useful small ways?
Specific examples/networks that we could not guess, not buzz words.. robust, modular, evolvable...
Invent fitness: generic, common to multi-phyla
Parameter dependencies: gradient descent, good enough local minima
Minimal model, geometric (dynam. sys.) of time and space process.(vs redundant genetic pathways, used variously in different species)
Experimental: High level: interconvert static to sliding morphogens.
Low level: functional form to fit C(x,t).
Theory: Evolved models ≠ 19th C applied math.
