Compu&ng with Chemical Reac&on Networks. Nikhil Gopalkrishnan

Transcription

1 Compu&ng with Chemical Reac&on Networks Nikhil Gopalkrishnan

2 The programming language of chemical kinetics Use the language of coupled chemical reactions prescriptively as a programming language for engineering new systems (rather than descriptively as a modeling language for existing systems) These gloves came free with my toilet brush! (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 2

3 Chemical Reaction Networks (CRN) syntax: we use only stochastic CRNs in this talk 4 (From presenta&on of Chen, Doty, Soloveichik, Determinis&c

4 (From presenta&on of Chen, Doty, Soloveichik, Determinis&c

5 (From presenta&on of Chen, Doty, Soloveichik, Determinis&c

6 Cells are smart: controlled by signaling and regulatory networks Human neutrophil chasing a bacterium through red blood cells source: David Rogers, Vanderbilt University Want to understand principles of chemical computation Engineer embedded controllers for biochemical systems, wet robots, smart drugs, etc. (From presenta&on of Chen, Doty, Soloveichik, Determinis&c

7 Are CRNs an implementable programming language? I don't believe that every crazy CRN you write down actually describes real chemicals! Response to objection: Soloveichik, Seelig, Winfree [PNAS 2010] found a physical implementation (high-accuracy approximation) of any CRN, using nucleic-acid strand displacement cascades (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 7

8 Formal Defini&on of Discrete (Stochas&c) CRN Model Finite set of species {X, Y, Z, } A state is a nonnegative integer vector c indicating the count (number of molecules) of each species: write counts as # c X, # c Y, Finite set of reactions: e.g. X W + Y + Z A + B C (assume all rate constants are 1, and all reactions are unimolecular or bimolecular) (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 8

9 Stochas(c Chemical Reac(on Network Stochastic Chemical Reaction Network(SCRN): defined as a finite set of d reactions acting on a finite number m of species. Each reaction α is defined as: A vector of nonnegative integers specifying the stoichiometry of the reactants, r α = (r α,1,..., r α,m ), Together with another vector of nonnegative integers specifying the stoichiometry of the products, p α = (p α, 1,..., p α,m ). The stoichiometry is the nonnegative number of copies of each species required for the reaction to take place, or 9 produced when the reaction does take place.

10 Stochas(c Chemical Reac(on Network We will use capital letters to refer to various species and we will use standard chemical notation to describe reactions. Example: The reaction A + D A + 2E consumes 1 molecule of species A and 1 molecule of species D and produces 1 molecule of species A and 2 molecules of species E. In this reaction, A acts catalytically because it must be present for the reaction to occur, but its number is unchanged when the reaction does occur. 10

11 Stochas(c Chemical Reac(on Network The state of the network: defined as a vector of nonnegative integers specifying the quantities present of each species, A = (q 1,..., q m ). A reaction is possible in state A: Only if there are enough reactants present, that is, i, q i r α,i. When reaction α occurs in state A, The reactant molecules are used up and the products are produced. The new state is: B = A α = (q 1 r α,1 + p α,1,..., q m r α,m + p α,m ). 11

12 Stochas(c Chemical Reac(on Network Reaction Notation: We write A C B if there is some reaction in the Stochastic Chemical Reaction Network C that can change A to B. We write C* for the reflexive transitive closure of C. Probabilistic Reactions: We write Pr[A C B] to indicate the probability that, given that the state is initially A, the next reaction will transition to the state to B. Pr[A B] refers to the probability that at some time in the future, the system is in state B. 12

13 Discrete (Stochas&c) CRN Model System evolves via a continuous time Poisson process: Reaction j A #A Propensity ρ j A + B (1/v) #A #B v = volume A + A (1/v) #A (#A 1) / 2 Time until next reaction is exponential random variable with rate Σ j ρ j (and expected value 1 / Σ j ρ j ) Probability that the next reaction is j* is ρ j* / Σ j ρ j 13

14 Stochas(c Chemical Reac(on Networks Papers Computation With Finite Stochastic Chemical Reaction Networks (Soloveichik,Cook, Winfree, Bruck, 1985) DNA as a universal substrate for chemical kinetics(soloveichik, Seelig, Winfree, PNAS2010) Programmability of Chemical Reaction Networks(Chen, Doty, Soloveichik) 14

15 Simula&on of Boolean circuits using CRNs Programmability of Chemical Reac(on Networks(Chen, Doty, Soloveichik) Given: Boolean Circuit: The Boolean Circuit suffices to use only one type of Boolean opera&on: NAND: Construct: Simula(ng CRN: It determinis&cally computes the same func&on as the given Boolean circuit, despite the uncontrollable order in which reac&ons occur. The presence of a single A i molecule represents that x i = 0, the presence of a single B i molecule represents that x i = 1, and the presence of neither indicates that x i has not yet been computed. If one starts with a single A or B molecule for each input variable, then with probability 1 the correct species will be eventually produced for each output variable.

16 Computa&ons using CRNs Programmability of Chemical Reac(on Networks(Chen, Doty, Soloveichik) Four representa(ons of the same computa(on. Star&ng with 1 A and n C s, the maximum number of D s that can be produced is 2n. (a) A Stochas(c Chemical Reac(on Network: (b) A Petri net: Each circle corresponds to a place (a molecular species), and each black bar corresponds to a transi,on (a reac(on). (c) A Vector Addi(on System: Note that dimensions F and G must be added to the Vector Addi(on System to capture the two reac(ons that are catalyzed by A and B. (d) A Fractran program: The numerators correspond to the reac(on products, and the denominators correspond to the reactants. The first seven prime numbers are used here in correspondence to the lemers A through G in the other examples. As in the previous example, F (13) and G (17) must be introduced here to avoid unreduced frac(ons for the catalyzed reac(ons

17 Determinis&c Func&on Computa&on with Chemical Reac&on Networks Ho-Lin Chen 1, David Doty 2, David Soloveichik 3 1 National Taiwan University 2 Caltech 3 UCSF 17

18 Determinis&c func&on computa&on with CRNs (defini&on) task: compute function z = f(x) (x N k, z N l ) initial state: input counts X 1, X 2,, X k (and fixed counts of non-input species) output: counts of Z 1, Z 2,, Z l output-stable state: all states reachable from it have same counts of Z 1, Z 2,, Z l deterministic computation: a correct output-stable state always reached in the limit t (infinitely often reachable states are infinitely often reached) (From presenta&on of Chen, Doty, Soloveichik, Determinis&c

19 Example 1 of Determinis&c func&on computa&on with CRNs z f(x) = 2x start with x (input amount) of X X Z + Z (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 19

20 Example 2 of Determinis&c func&on computa&on with CRNs f(x 1,x 2 ) = if x 1 > x 2 then y = 1 else y = 0 start with 1 N and input amounts of X 1,X 2 X 1 + N Y X 2 + Y N (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 20

21 Example 3 of Determinis&c func&on computa&on with CRNs f(x 1,x 2 ) = max {x 1,x 2 } start with input amounts of X 1,X 2 Z X 1 Z 1 + Z X 2 Z 2 + Z Z 1 + Z 2 K K + Z Ø (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 21

22 Other func&ons? f(x) = x/2? f(x) = x 2? f(x 1,x 2 ) = x 1 x 2? f(x) = 2 x? (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 22

23 Main result Theorem: Functions f: N k N l deterministically computable by CRNs are precisely those with a semilinear graph. graph(f) = { (x,z) N k+l f(x)=z } A N k+l is linear if there are vectors b, u 1,, u p so that A = { b + n 1 u n p u p n 1,...,n p N } A is semilinear if it is a finite union of linear sets. Intuitively, semilinear functions are piecewise linear functions with a finite number of pieces (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 23

24 Non-semilinear examples Others: f(x) = x 2 f(x) = 2 x (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 24

25 How do we show this? Theorem [Angluin, Aspnes, Eisenstat, PODC 2006]: The predicates decidable by CRNs are precisely the semilinear predicates. We connect computation of functions (integer output) to computation of predicates (YES/NO output) (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 25

26 Determinis&c predicate computa&on with stochas&c CRNs: task: decide predicate b = φ(x) (x N k, b {yes,no}) initial state: input counts X 1, X 2,, X k (and fixed counts of non-input species) output: either #Y > 0 and #N = 0 (yes) or #Y = 0 and #N > 0 (no) output-stable state: all states reachable from it have same yes/no answer set decided by CRN: S yes = { x N k φ(x) = yes } (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 26

27 Theorem [Angluin, Aspnes, Eisenstat, PODC 2006]: The sets decidable by CRNs are precisely the semilinear sets. Two direc(ons of proof Only semilinear functions can be computed: f computed by CRN C graph(f) decided by CRN D All semilinear functions can be computed: graph(f) decided by CRN D f computed by CRN C (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 27

28 f computed by CRN C graph(f) decided by CRN D Want to decide, given input (x,z), is f(x) = z? Keep track of total number of Z's ever produced or consumed: A + B Z + W A + Z B becomes A + B Z + W + Z P becomes A + Z B + Z C Initial state has z copies of Z C Z P + Z C Y Z P + Y Z P + N Y + N Y Z C + Y Z C + N If Z P or Z C are left over, change answer to NO Eventually all Z P and Z C go away (if equal) or one is left over (if unequal) If neither is left over, change answer to YES (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 28

29 graph(f) decided by CRN D f computed by CRN C Want: given x copies of X, produce f(x) copies of Z If graph(f) = { (x,z) N 2 f(x) = z } is semilinear, then so is the set F diff = { (x,z P,z C ) N 3 f(x) = z P z C } So some CRN D diff decides F diff Start with 0 of Z, Z P, Z C, and add to D diff the reactions N only present when D diff thinks answer is NO N N + Z P + Z N + Z N + Z C (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 29

30 CRN Simula(ons of: Minsky s register machine (RM) Finite state machine with a fixed number of registers Each register can store a non-nega&ve integer Inc(i,r,j) : Increment register r and move from state i to j Dec(I,r,j,k): Decrement register r if r > 0 and move from state i to j; else move to state k

31 Minsky s register machine (RM) Finite state machine with a fixed number of registers Each register can store a non-nega&ve integer Inc(i,r,j) : Increment register r and move from state i to j Dec(I,r,j,k): Decrement register r if r > 0 and move from state i to j; else move to state k A register machine comparing the value of register R 1 to R 2. If R 1 R 2, then it outputs 1 in register R 3. If R 1 > R 2 then it outputs 2 in register R 3. The start state is indicated with start and the hal&ng states are those without outgoing arrows

32 Computa&ons using CRNs Programmability of Chemical Reac(on Networks(Chen, Doty, Soloveichik) Simula(ng a register machine: (a) The communica(on between the clock and the register logic modules is through single molecules of species C and T. (b) The clock module is responsible for producing a C molecule once every so oden. The clock module is designed so that the length of &me between receiving a T and producing a C slowly increases throughout the computa&on, thus slowing down the register logic module to help it avoid error. The more A s there are, the longer the delay. The clock starts out with n 0 A s and one each of B, Bʹ, and Bʹʹ and T. Every clock cycle not only produces a C, but increases the number of A s by one. Thus, at the beginning of the kth cycle, there are n = k + n 0 molecules of A. (c) The register logic module simulates the register machine state transi(ons. The register logic module starts out with quan&&es of molec ules of R i indica&ng the star&ng value of register i, and a single molecule of species S a where a is the start state of the register machine. Note that at all &mes the en&re system contains at most a single molecule of any species other than the A and R i species. All rate constants are 1 (The construc&on will work with any rate constants)

33 Computa&ons using CRNs Programmability of Chemical Reac(on Networks(Chen, Doty, Soloveichik) Simula(ng a register machine: The state diagram for a single decrement opera&on when: There are n A s and the register to be decremented holds the value 1, and The corresponding system of differen&al equa&ons governing the instantaneous probabili&es of being in a given state. The numbers on the arrows are the transi&on rates. The instantaneous probability of being in state T is s, in state T ʹ is sʹ, and in state T ʹʹ is sʹʹ. The instantaneous probability of being in the error-possible state is p and the probability of being in the no-error state is q

34 What if we allow error? Any function computable by an algorithm is computable by a randomized CRN with arbitrarily small positive probability of error. [Soloveichik, Cook, Winfree, Bruck, Natural Computing 2008] [Angluin, Aspnes, Eisenstat, Distributed Computing 2006] Moral: disallowing error hurts chemical algorithms much more than it hurts conventional algorithms randomized finite-state algorithms deterministic finite-state algorithms finite-state computable deterministic CRNs semilinear deterministic poly-time (P) algorithms randomized poly-time (BPP) algorithms polynomial-time computable exponential-time computable randomized algorithms deterministic algorithms randomized CRNs algorithmically computable function's computational complexity (From presenta&on of Chen, Doty, Soloveichik, Determinis&c 34 18

35 Computa(on With Finite Stochas(c Chemical Reac(on Networks (Soloveichik,Cook, Winfree, Bruck, 1985) SCRNs are Turing universal and thus can compute any computable func&on without error, reac4on. assuming fast reac4ons are guaranteed to occur before any slow SCRNs can compute any computable func(on with probability of error less than ε for any ε > 0, but for ε = 0 universal computa4on is impossible. SCRNs are NOT capable of universal computa(on with any fixed bounded probability of success, if each reac4on s probability of occuring depends only on what reac4ons are possible (but not on the concentra4ons). SCRNs are capable of compu(ng exactly the class of primi(ve recursive func(ons without error, if we take the result of the longest possible sequence of reac4ons as the answer. The (me and space requirements for Stochas&c Chemical Reac&on Networks doing computa&on, compared to a Turing Machine, are a simple polynomial slowdown in 4me, but an exponen4al increase in space.

36 Computa(on With Finite Stochas(c Chemical Reac(on Networks (Soloveichik,Cook, Winfree, Bruck, 1985) Turing-universal computa&on using molecular counts Fast and reliable Small number of dis&nct molecular species Probability of error can be made arbitrarily small (> 0) by increasing molecular counts

37 Naive Simula&on of Register Machine (RM) by Stochas&c CRN Error Worse when M r = 1 Error per step = K 2 / (K 1 /v+k 2 ) K 1 = rate constant for dec 1 K 2 = rate constant for dec 2

38 Improved Bounded RM Simula&on Rxn Logical function (inc) (dec 1 ) (dec 2 ) Bounded RM simula(on. Species C (#C = 1) acts a dummy catalyst to ensure that all reac&ons are bimolecular, simplifying the analysis of how the simula&on scales with the volume. Ini(al molecular counts are: if i is the start state then #S = 1, i #S = 0 for j i, and j #M is the ini&al value of register r. r

39 Improved Bounded RM Simula&on Rxn Catalysts Clock module for the RM simula(on: Dummy Catalyst A : acts as a dummy catalyst to ensure that all reac&ons in the clock module are bimolecular and thus scale equivalently with the volume, ensuring that the error probability is independent of the volume. Clock Module: The clock module maintains the average concentra&on of C i at approximately (#A ) i /(#A) i 1. Ini(al molecular counts are: #C i = 1, and #C 1 = = #C i 1 = 0. For the RM simula&on #A = 1, and #A = Θ(1/ε i/(i 1) ).