The Cell Cycle Switch Computes Approximate Majority

Transcription

1 The Cell Cycle Switch Computes Approximate Majority Luca Cardelli, Microsoft Research & Oxford University Joint work with Attila Csikász-Nagy, Fondazione Edmund Mach & King s College London Middlesex Algorithms Day,

2 Outline Algorithms and Dynamical Systems Networks and Morphisms Kinetic Emulation Network Zoos Conclusions 2

3 Algorithms and Dynamical Systems

4 Biological Networks activation inhibition Mutual Inhibition & Self Activation Mutual Inhibition & Mutual Anti-activation Something Mysterious Something Complicated MI SI CC NCC Cell cycle transitions Septation Initiation The G 2 /M cell cycle switch The new cell cycle switch Polarity establishment Gene networks SIN inhibiting Byr4, absence of SIN activating Byr4 s t z r x 4

5 Consensus Algorithms catalysis Approximate Majority (AM) Epigenetic Switch Two initial populations: some x 0 + some x 2 One final population: all x 0 or all x 2 One intermediate population: x 1 (undecided) x 0 x 1 x 2 x 0 + x 2 x 2 + x 1 x 2 + x 0 x 0 + x 1 x 1 + x 0 x 0 + x 0 x 1 + x 2 x 2 + x 2 x 0 =x 2 Worst-case scenario, starting with x 0 =x 2, x 1 =0: Provably fast: O(log n) and robust to perturbations 5

6 Influence Nodes inhibition = inhibit x high low x is high x is low activation inhibition catalysis Approximate Majority activation Usually modeled by sigmoid (e.g. Hill or Reinitz) functions activate x We model them by 4 mass action reactions over 3 species x 0, x 1, x 2 They actually implement a Hill function of coefficient 2: = r 21 = 0.1 r 10 = 10.0 r 01 = 0.1 r 12 = 10.0 AM 6

7 In Previous Work activation inhibition catalysis The classical Cell Cycle Switch CC approximates AM performance (a bad switch) Stabilization Speed x s 150 Steady State Stimulus- Response 0 0 x s 0 0 DC AM CC t s sx s time stimulus t p Pr(x p t p ) x p 0 x p 0 sx p Pr(x p sx p ) CC converges in O(log n) time (like AM) (but 2x slower than AM, and does not fully switch) Symmetrical initial conditions (x 0 =x 1 =x 2 ) Black lines: high-count stochastic simulation traces Color: full probability distribution of low-count system Hor axis is time. AM shows hysteresis (like CC) Black lines: deterministic ODE bifurcation diagrams Red lines: medium-count stochastic simulations Color: full probability distribution of low-count system Hor axis is stimulus pushing towards x 0 against fixed bias. There is an obvious bug in CC performance: let s fix it! 7

8 In Previous Work But GW is better! Fully switchable, just as fast as AM GW emulatesam GW That same week: The Greatwallloop is a necessary component of the switch So, nature fixed CC! 8

9 Networks and Morphisms

10 A Theory of Network Emulation (with thanks to David Soloveichik) So far, evidence is empirical Simulations based on a choice of parameters But indeed... We can show that, GW, NCC, etc. are exactly and alwaysas good as AM Where exactly means numerically as good, not just in the same complexity class And alwaysmeans for any choice of rates and initial conditions 10

11 Network Emulation: MI emulates AM For any rates and initial conditions of AM, we can find some rates and initial conditions of MIsuch that the (6) trajectories of MIretrace those (3) of AM: MI ~y,z x AM (3 species) initialize: z = x ~y = x (y 2 = x 0 y 1 = x 1 y 0 = x 0 ) (6 species on 3 trajectories) (3 species on 3 trajectories) 11

12 Emulation is a Network Morphism A mapping of species and reactions any initial conditions AM homomorphic mapping initial conditions: z 0 = y 2 = x 0 z 1 = y 1 = x 1 z 2 = y 0 = x 2 MI less trivial than you might think: it need not preserve the out-degree of a node! 12

13 Network Emulation: NCC emulates MI For anyrates and initial conditions of MIwe can find somerates and initial conditions of NCC such that the (18) trajectories of NCC retrace those (6) of MI NCC MI (3 species each) NCC z,r,p z y,q,s y MI (18 species on 6 trajectories) Why does this work so well? (6 species on 6 trajectories) initialize z,r,p = z y,q,s = y 13

14 Kinetic Emulation

15 When can a Network Emulate Another? What kind of morphisms guarantee emulation? do they preserve network structure? do they preserve stoichiometry? 15

16 Chemical Reaction Networks A CRN is a pair, where = {,, } = {,, } a finite set of species a finite set of reactions = {!, ", #} =! " #, Reactions = with stoichiometric numbers, N = 2! + "! + 3# ' = 2, ( = 1, * = 0 ' = 1, ( = 0, * = 3! " #, The stoichiometryof in is: (, ) = (, ) = ( )!, = 1!, = net stoichiometry (instantaneous) stoichiometry 16

17 CRN Morphisms A CRN morphism from, to (-,.) written /, (-,.) Mappings (symmetries) between two networks is a pair of maps / = / 0, / R a species map / 0 - a reaction map / R. (sometimes omitting the subscripts on /) We are interested in morphisms that are not injective, that represent refinements of simpler networks 17

18 3 Key Morphisms A morphism /, -,. is a CRN homomorphism if / R is determined by / 0 : / R = / 0 / 0 7, 78 are the respective stoichiometric matrices :, :8 are the respective reactant matrices 4 5, 4 9 are the characteristic 0-1 matrices of / 0, / R 4 5, = 1 if / 0 = else = a CRN reactant morphism if / R is determined by / 0 on reactants.., 3: / R = / 0. 3 a CRN stoichiomorphism if: def : = : = / 0 = Σ 0 => ( ) 18

19 Checking the Stoichiomorphism Condition / MI AM... Σ P => P, = /, MI mi 2 mi 3 AM am 2 am 3 mi 1 mi 0 am 1 am 0 /(J A ) /? K/ A mi 6 mi 7 J A, /@ A + J A, /@ B = 1 = I D, K/ A mi 5 mi 4 All unit rates (sufficient because of another theorem) This is both a homomorphism and a stoichiomorphism /? ( ) /@ A, /@ B /@, /@ C /@ D, /@ E /@ F, /@ G H A I A H I H D I D J A I D J I J D I A K/ A K/ K/ D K/ F AM 19

20 CRN Kinetics A stateof a CRN, is a R R T a vector of concentrations for each species The differential systemof a CRN (, ), U R T R R R U(R)() ` Given by the law of mass action: U(R)() = Σ PV(W X Y) Z (, ) Π R ] W U(R)()gives the instantaneous change of concentration of a species in a given state sum over all reactions of the stoichiometry of species in reaction times the product of reagent concentrations according to their stoichiometric numbers Usually written as a system of coupled concentration ^R ODEs, integrated over time: ] = U(R)() ^_ 20

21 Kinetic Emulation,,b A morphism /, (-,.)is a CRN emulation if for the respective differential systems U, Ū, R8 R T - : U R8 / = Ū R8 / R R T / R8 R T U if the derivative of (in state R8 /) equals the derivative of / (in state R8) That is:. U R8 / = Ū R8 (/ ) R = R8 / R, ` R8 () U R = U. R8 (/ ) R8 /() (-,.) ` if we startthe two systems in states R = R8 / (which is a copyof R8according to /) and R8 resp., for each the solutions are equal and the derivatives are equal, hence they will have identical trajectories by determinism 21

22 Emulation Theorem Theorem: If /, -,. is a CRN reactant morphism and stoichiomorphism then it is a CRN emulation thus, for any initial conditionsof -,. we can match trajectories reactant morphism stoichiomorphism : = : = AM AM emulation U R8 / = U. R8 / MI MI N.B. homomorphism implies reactant morphism, implies 4 56 : = :

23 Change of Rates Theorem A change of rates for, is morphism c,, such that c()is the identity and c,, =,, e. Theorem: If /, (-,.)is a stoichiomorphism, then for anychange of rates c of (-,.)there is a change of rates cof, such that c / c? is a stoichiomorphism. a morphism that modifies rates only thus, for any ratesof -,. we can match trajectories In fact, c changes rates by the ratio with which c changes rates: c,, =,,. f. where /,, = (3, 3,.)and c 3, 3,. = (3, 3,. ). 23

24 Network Zoos

25 Approximate Majority Emulation Zoo p r q s DN z,~y z s,~r y z,~y z s,~r y x z s,~r y SI z,~y x r,~s x CCr NCC q s p r q s p r QI z,~y x z,r z y,s y CCr r,~s x z,~y x AM r,~s x r,~s x SCr s y r z r z MI GW s y SCr ( homomorphism and stoichiomorphism (transitive)) 25

26 Emulations Compose: NCC emulates AM The (18) trajectories NCCcan always retrace those (3) of AM z,r,p z y,q,s y z,~y x NCC z,r,p x ~y,~q,~s x AM (18 species on 3 trajectories) (3 species on 3 trajectories) 26

27 Emulation in Context / MI AMis an emulation: it maps H I and ~h I We can replace AM with MI in a context.the mapping / tells us how to wire MI to obtain an overall emulation: AM-AM Oscillator Each influence crossing the dashed lines into I is replaced by a similar influence into both H and ~h. The latter is the same as an opposite influence into h (shown). AM-MI Oscillator Each influence crossing the dashed lines out of I is replaced by a similar influence from the same side of either Hor ~h. The latter is the same as a similar influence from the opposite side of h (shown), and the same as an opposite influence from the same side of h. 27

28 Another Zoo 28

29 Network Perturbations Network Normal Behavior Removing each link in turn dead never dead on average A complex but robust implementation of the simple network 29

30 Conclusions

31 Interpretations of Stoichiomorphism Explanation of network structure E.g. we know that the main function of Delta-Notch is to stabilize the system in one of two states. AM is the quintessential network that embodies fast robust bistability. The stoichiomorphism from Delta-Notch to AM explains what Delta-Notch(normally) does, and exactly how well it can do it. Robust implementation of simpler function Redundant symmetries are implicit in the stoichiomorphism relationships Neutral paths in network space (evolution) If an evolutionary event happens to be a stoichiomorphism, or close to it, it will not be immediately selected against, because it is kinetically neutral. This allows the network to increase its complexity without kinetic penalty. Later, the extra degrees of freedom can lead to kinetic differentiation. But meanwhile, the organism can explore variations of network structure. Network implementation (not abstraction!) Stoichiomorphisms are not about abstraction / coarse-graining that preserve behavior, on the contrary, they are about refinement / fine-graining that preserve behavior. They describe implementationsof abstract networks, where the abstract networks themselves may not be (biologically) implementable because of excessive demands on species interactions. 31

32 Nature likes a good algorithm First part Second part Approximate default rates and initial conditions CC Exact any rates and initial conditions CCr These additional feedbacks do exist in real cell cycles (via indirections) The cell cycle switch can exactly emulate AM NCC MI AM 32

33 In separate work... We produced a chemical implementation of AM using DNA gates I.e., a synthetic reimplementation of the central cell-cycle switch. 33