The Effect of Randomness in Chemical Reactions on Decision-Making: Two Short Stories

Transcription

1 The Effect of Randomness in Chemical Reactions on Decision-Making: Two Short Stories J. P. Keener Department of Mathematics University of Utah Length Regulation of Flagellar Hooks p.1/46

2 The Fundamental Problem All living organisms make decisions (when to divide, when to differentiate, when to destroy, when to repair, when to grow, when to die) Question 1: What is the chemistry of decision making? How is information about size, age, resource availability, etc. transduced into chemistry? Question 2: When the organism or organelle is small, the number of molecules involved in the decision making process is necessarily small. How does the resultant randomness affect the reliability of the decision? To be or not or not to be, that is the question. p.2/46

3 Story 1: Length Regulation Observation: Every structure which could in principle be of any size but instead is found in a narrow distribution of sizes must be endowed with a size control mechanism. Question 1: How are the specifications for the size of a structure set? Question 2: How are the decisions regarding manufacture made? (When, what and how much to build?) The first of many slides p.3/46

4 Example: Cilia Hair cell Steriocilia bronchial cilia The first of many slides p.4/46

5 Example:Tails phage lambda chlamydomonas The third of many slides p.5/46

6 Injectosomes s Type III Secretion System - Injectosome The fourth of many slides p.6/46

7 Specific Example: Salmonella Length Regulation of Flagellar Hooks p.7/46

8 Control of Flagellar Growth The motor is built in a precise step-by-step fashion. Step 1: Basal Body Step 2: Hook (FlgE secretion) Step 3:Filament (FliC secretion) Basal Body Length Regulation of Flagellar Hooks p.8/46

9 Control of Flagellar Growth The motor is built in a precise step-by-step fashion. Step 1: Basal Body Step 2: Hook (FlgE secretion) Step 3: Filament (FliC secretion) FlgE Hook Basal Body Length Regulation of Flagellar Hooks p.8/46

10 Control of Flagellar Growth The motor is built in a precise step-by-step fashion. FliC Step 1: Basal Body Step 2: Hook (FlgE secretion) Step 3: Filament (FliC secretion) Hook FlgE Filament Hook filament junction Basal Body Click to see movies Length Regulation of Flagellar Hooks p.8/46

11 Control of Flagellar Growth The motor is built in a precise step-by-step fashion. FliC Step 1: Basal Body Step 2: Hook (FlgE secretion) Step 3: Filament (FliC secretion) Hook FlgE Filament Hook filament junction Basal Body Click to see movies How are the switches between steps coordinated? How is the hook length regulated (55 ±6 nm)? Length Regulation of Flagellar Hooks p.8/46

12 Proteins of Flagellar Assembly Length Regulation of Flagellar Hooks p.9/46

13 Hook Length Regulation Hook is built by FlgE secretion. Length Regulation of Flagellar Hooks p.1/46

14 Hook Length Regulation Hook is built by FlgE secretion. FliK is the "hook length regulatory protein". Length Regulation of Flagellar Hooks p.1/46

15 Hook Length Regulation Hook is built by FlgE secretion. FliK is the "hook length regulatory protein". FliK is secreted only during hook production. Length Regulation of Flagellar Hooks p.1/46

16 Hook Length Regulation Hook is built by FlgE secretion. FliK is the "hook length regulatory protein". FliK is secreted only during hook production. Mutants of FliK produce long hooks; overproduction of FliK gives shorter hooks. Length Regulation of Flagellar Hooks p.1/46

17 Hook Length Regulation Hook is built by FlgE secretion. FliK is the "hook length regulatory protein". FliK is secreted only during hook production. Mutants of FliK produce long hooks; overproduction of FliK gives shorter hooks. Lengthening FliK gives longer hooks. Length Regulation of Flagellar Hooks p.1/46

18 Hook Length Regulation Hook is built by FlgE secretion. FliK is the "hook length regulatory protein". FliK is secreted only during hook production. Mutants of FliK produce long hooks; overproduction of FliK gives shorter hooks. Lengthening FliK gives longer hooks. 5-1 molecules of FliK are secreted per hook ( molecules of FlgE). Length Regulation of Flagellar Hooks p.1/46

19 Hook Length Data Mutants of FliK produce long hooks; overproduction of FliK gives shorter hooks: Number of hooks 15 1 Number of hooks 15 1 Number of hooks (a) L(nm) (b) L(nm) (c) L(nm) Wild type Overexpressed Underexpressed (M = 55nm) (M = 47nm) (M = 76nm) Length Regulation of Flagellar Hooks p.11/46

20 The Secretion Machinery Secreted molecules are chaperoned to prevent folding. CM MS ring C ring membrane components FlhA FlhB N FliJ FliH FliI C Step 1 Length Regulation of Flagellar Hooks p.12/46

21 The Secretion Machinery Secreted molecules are chaperoned to prevent folding. Insertion is facilitated by hydrolysis of ATP. CM MS ring C ring membrane components FlhA FlhB N FliJ FliH FliI Step 2 C Length Regulation of Flagellar Hooks p.12/46

22 The Secretion Machinery Secreted molecules are chaperoned to prevent folding. Insertion is facilitated by hydrolysis of ATP. FlhB is the gatekeeper recognizing the N terminus of secretants. CM MS ring C ring membrane components N FlhA FlhB FliJ FliH FliI ATP C ADP+Pi Step 3 Length Regulation of Flagellar Hooks p.12/46

23 The Secretion Machinery Secreted molecules are chaperoned to prevent folding. N Insertion is facilitated by hydrolysis of ATP. FlhB is the gatekeeper recognizing the N terminus of secretants. Once inside, molecular movement is by diffusion. CM MS ring C ring FliJ membrane components FlhA C FlhB Step 4 FliI FliH Length Regulation of Flagellar Hooks p.12/46

24 Secretion Control Secretion is regulated by FlhB During hook formation, only FlgE and FliK can be secreted. After hook is complete, FlgE and FliK are no longer secreted, but other molecules can be secreted (those needed for filament growth.) The switch occurs when the C-terminus of FlhB is cleaved by FlK. Question: Why is the switch in FlhB length dependent? Length Regulation of Flagellar Hooks p.13/46

25 Hypothesis: How Hook Length is determined The Infrequent Molecular Ruler Mechanism. FliK is secreted once in a while to test the length of the hook. The probability of FlhB cleavage is length dependent. Length Regulation of Flagellar Hooks p.14/46

26 Binding Probability Suppose the probability of FlhB cleavage by FliK is a function of length P c (L). Then, the probability of cleavage on or before time t, P(t), is determined by dp dt = αr(l)p c(l)(1 P) where r(l) is the secretion rate, α is the fraction of secreted molecules that are FliK, and dl dt = βr(l) where β = 1 α fraction of secreted FlgE molecules, length increment per FlgE molecule. Length Regulation of Flagellar Hooks p.15/46

27 Binding Probability It follows that or dp dl = α β P c(l)(1 P) ln(1 P(L)) = κ L P c (L)dL Remark: The only difference between mutant strains should be in the parameter κ = α β. Length Regulation of Flagellar Hooks p.16/46

28 Check the Data Number of hooks 15 1 Number of hooks 15 1 Number of hooks (a) L(nm) (b) L(nm) (c) L(nm) Wild type Overexpressed Underexpressed 1.9 Overexpressed WT Underexpressed P(L) Hook Length (nm) Length Regulation of Flagellar Hooks p.17/46

29 Check the Data ln(1 P) Overexpressed WT Underexpressed Hook Length (nm) κ ln(1 P) κ =.9 κ = 1 κ = Hook Length (nm) ln(1 P(L)) = κ L P c (L)dL? Length Regulation of Flagellar Hooks p.18/46

30 Use the Data to estimate P c (L): P c (L) Hook Length (nm) Hook Length (nm) Length Regulation of Flagellar Hooks p.19/46

31 Test #2: 3 Cultures number of hooks 1 5 number of hooks hook length (nm) hook length (nm) 1: WT 2: No FliK number of hooks hook length (nm) 3: FliK induced at 45 min Question: Can 3 be predicted from 1 and 2? Length Regulation of Flagellar Hooks p.2/46

32 Test #2 Suppose that FliK becomes available only after the hook is length L. Then, dp dl = βp c(l)(1 P). with P(L L ) = so that P(L L ) = 1 exp( κ L Let s see how this formula can be used: L P c (L)dL). Length Regulation of Flagellar Hooks p.21/46

33 Step 1: Analysis of WT Data 1) Determine P c (L) from WT data number of hooks 1 5 P(L) hook length (nm) Hook Length (nm) ln(1 P) 4 3 β P c (L) (nm 1 ) Hook Length (nm) Hook Length (nm) Length Regulation of Flagellar Hooks p.22/46

34 Step 2: Analysis of Polyhook Data 2) Determine distribution of polyhooks P p (L) (hooks grown with no hook length control) number of hooks P p (L) hook length (nm) hook length Length Regulation of Flagellar Hooks p.23/46

35 Step 3: Predict Lengths after FliK induction P i (L) = P(L )P p (L )+ L where P(L L ) = 1 exp( κ L L P c (L)dL). P(L L )p p (L +L )dl. number of hooks hook length (nm) P i (L) hook length (nm) Length Regulation of Flagellar Hooks p.24/46

36 Hypothesis: How Hook Length is determined The Infrequent Molecular Ruler Mechanism. The probability of FlhB cleavage is length dependent. What is the mechanism that determines P c (L)? Length Regulation of Flagellar Hooks p.25/46

37 Secretion Model Hypothesis: FliK binds to FlhB during translocation to cause switching of secretion target by cleaving a recognition sequence. FliK molecules move through the growing tube by diffusion. L x Length Regulation of Flagellar Hooks p.26/46

38 Secretion Model Hypothesis: FliK binds to FlhB during translocation to cause switching of secretion target by cleaving a recognition sequence. FliK molecules move through the growing tube by diffusion. They remain unfolded before and during secretion, but begin to fold as they exit the tube. L x Length Regulation of Flagellar Hooks p.26/46

39 Secretion Model Hypothesis: FliK binds to FlhB during translocation to cause switching of secretion target by cleaving a recognition sequence. FliK molecules move through the growing tube by diffusion. They remain unfolded before and during secretion, but begin to fold as they exit the tube. Folding on exit prevents back diffusion, giving a brownian ratchet effect. L x Length Regulation of Flagellar Hooks p.26/46

40 Secretion Model Hypothesis: FliK binds to FlhB during translocation to cause switching of secretion target by cleaving a recognition sequence. FliK molecules move through the growing tube by diffusion. They remain unfolded before and during secretion, but begin to fold as they exit the tube. Folding on exit prevents back diffusion, giving a brownian ratchet effect. For short hooks, folding prevents FlhB cleavage. L x Length Regulation of Flagellar Hooks p.26/46

41 Secretion Model Hypothesis: FliK binds to FlhB during translocation to cause switching of secretion target by cleaving a recognition sequence. FliK molecules move through the growing tube by diffusion. They remain unfolded before and during secretion, but begin to fold as they exit the tube. Folding on exit prevents back diffusion, giving a brownian ratchet effect. For short hooks, folding prevents FlhB cleavage. For long hooks, movement solely by diffusion allows more time for cleavage. L x Length Regulation of Flagellar Hooks p.26/46

42 Stochastic Model Follow the position x(t) of the C-terminus using the langevin equation L νdx = F(x)dt+ 2k b TνdW, where F(x) represents the folding force acting on the unfolded FliK molecule, W(t) is brownian white noise. x Length Regulation of Flagellar Hooks p.27/46

43 Fokker-Planck Equation Let P(x,t) be the probability density of being at position x at time t with FlhB uncleaved, and Q(t) be the probability of being cleaved by time t. Then and P t = x (F(x)P)+D 2 P x 2 g(x)p, dq dt = b a g(x)p(x, t)dx. where g(x) is the rate of FlhB cleavage at position x. g(x) L x Length Regulation of Flagellar Hooks p.28/46

44 Probability of Cleavage To determine the probability of cleavage π c (x) starting from position x, solve D d2 π c dx 2 +F(x)dπ c dx g(x)π c = g(x) subject to π c(a) = and π c (b) =. Then P c (L) = π c (a). Pc(L) L Length Regulation of Flagellar Hooks p.29/46

45 Results CDF.5 CDF.5 CDF (a) L(nm) (b) L(nm) (c) L(nm) PDF.5 PDF.5 PDF (a) L(nm) (b) L(nm) (c) L(nm) Wild type Overexpressed Underexpressed The data are matched extremely well by the model predictions. Length Regulation of Flagellar Hooks p.3/46

46 Story 2: Action Potential initiation Deterministic models have been used with great success to describe many important decision making biological processes. However, the discrepancies between the model predictions and what actually happens are becoming more apparent. The cause of this discrepancy is often that the number of molecules involved is small and so stochastic behavior must be taken into account. Question 1: Why do deterministic and stochastic descriptions sometimes give different answers? Question 2: Specifically, what is the mechanism for the spontaneous (noise driven) firing of an action potential? Spontaneous Action Potentials p.31/46

47 Example: Keizer s Paradox Consider the autocatalytic reaction E +X k 1 k 2 2X, X k3 E Law of mass action: Let e = [E], x = [X], then dx dt = k 1ex k 2 x 2 k 3 x, e+x = E. This is the logistic equation with x = k 1E k 3 k 1 +k 2 solution, x = unstable steady solution. > stable steady Spontaneous Action Potentials p.32/46

48 Keizer s Paradox The stochastic version of this reaction: Let S j be the state in which there are j molecules of X and N j molecules of E, S j α j β j+1 S j+1, α j = j(n j) k 1 v, β j = jk 3 +j k 2 v. Notice that α = so that S is an absorbing state. Furthermore, the master equation for this reaction has a unique steady solution with p = 1 and p j = for all j. So, for all initial data, p(t) e as t i.e., j with probability 1 (i.e., extinction is certain), even though is an unstable rest point of the deterministic dynamics. Spontaneous Action Potentials p.33/46

49 Keizer s Paradox Visualized 1 4 # of x molecules time There appears to be (is) a slow manifold, extinction is a rare event. Spontaneous Action Potentials p.34/46

50 QSS Analysis Let ǫ = k 3v k 1 N 1. Then, we can write the master equation as dp dt = k 3p 1, dp dt = k 1N(A +ǫa 1 )p where A represents the reactions E +X k 1 k2 2X and A 1 represents the reactions X k 3 E. If p satisfies A p =, 1 T p = 1, set p = αp +ǫp 1 and project to find dα dt = k 3v1 T A 1 p α, an exponential process. Spontaneous Action Potentials p.35/46

51 Keizer s Paradox Resolved The nullspace of A is the binomial distribution ( ) N (k1 ) j k j 2 p j = (1+ k 1 k 2 ) N 1, j = 1,2, N, Consequently, the exit time is approximately τ e = 1 1 T A 1 p = k 2 (1+ k1 k 2 ) N 1 k 1 k 3 N which is exponentially large in N. Thus, even though extinction is certain, the expected extinction time is exponentially large. Spontaneous Action Potentials p.36/46

52 Example: Action Potential Dynamics I K I l I Na v Extracellular Space e C m I ion v = v v i e The Morris-Lecar equations Intracellular Space v i dv dt = g Nam (v)(v Na v)+g K w(v K v)+g l (v l v)+i where w satisfies dw dt = α w(v)(1 w) β w (v)w. Spontaneous Action Potentials p.37/46

53 Action Potential Dynamics This is a fast-slow system with excitable dynamics and threshold behavior. Question: How does noise affect the initiation of action potentials? Spontaneous Action Potentials p.38/46

54 Upstroke Dynamics To study upstroke dynamics, we hold w fixed (assuming w is a "slow" variable), and equation reduces to dv dt = g Nam (v)(v Na v)+g K w (v K v)+g l (v l v)+i G(v) v v = m (v)f(v) g(v) G (v) I = I I < I which is the bistable equation v [mv] One might conclude that spontaneous action potentials occur when noisy Na + currents push v out of one potential well into another. Spontaneous Action Potentials p.39/46

55 Discrete Ion Channels Since ion channels are noisy and discrete, a more appropriate model is dv dt = g n Na N (v m Na v)+g K M (v K v)+g l (v l v)+i where n and m are the number of open Na + and K + channels, respectively, and transition rates are voltage dependent. In particular, assume channels open and close via stochastic process C α(v) β(v) If there are N such channels, the ion channel state diagram is Nα (N 1)α S S 1 O. S 2 S N 1 α S N Spontaneous Action Potentials p.4/46

56 Stochastic Simulations V Na + Channels K + Channels w K + Channels V Observation: Action potentials are initiated by random fluctuations in K + channels, not Na + channels. Spontaneous Action Potentials p.41/46 Na +

57 The Chapman-Kolmogorov Equation Master equation for jump-velocity process p = {p(v, n, w, t)} where dv nw dt p t = ( ( dv ) nm ). p v dt + 1 ǫ A Na +p+a K +p, = g Na n N (v Na v)+g K w M (v K v)+g l (v l v)+i and A Na +, A K + are transition operators for Na + and K + channels. ǫ 1 since Na + channels are "fast". Goal: Exploit the fact that ǫ 1 and that A Na + represents a birth-death process. Spontaneous Action Potentials p.42/46

58 Analysis WKB: Seek equilibrium solution of the form p(v,w,n) = r(n v,w)exp( φ(v,w) ǫ substitute and find to leading order in ǫ an equation of the form ), H(v,w,p,q,) =, p = φ v, q = φ w a Hamilton-Jacobi equation. (H is too complicated to display.) This can be solved by method of characteristics: d dt ( p q dφ dt ) = p dv dt +qdw dt ) = ( H v H w along ( d dt v w ) = ( H p H q ) Spontaneous Action Potentials p.43/46

59 The Likelihood Surface The likelihood surface: (a "stochastic phase plane") Trajectories are most "most likely path" There is a caustic/ separatrix there is an approxiamte "saddle-node" (SN) Spontaneous Action Potentials p.44/46

60 Conclusion WKB analysis produces a maximum likelihood surface that has saddle-point behavior. Trajectories leading to action potentials may be Na + induced, if N is small, or K + -induced if M is small. The fast-slow analysis of the deterministic model is misleading with respect to stochastic, channel noise-induced action potentials. Spontaneous Action Potentials p.45/46

61 Thanks Thanks to Kelly Hughes (hooks) Berton Earnshaw (Channel Kinetics) Parker Childs (Keizer) Jay Newby (SAC s) Paul Bressloff (SAC s) NSF (funding!!) Spontaneous Action Potentials p.46/46