Spin coherent states and stochastic hybrid path integrals
SSpin coherent states and stochastic hybrid pathintegrals
Paul C. Bressloff Department of Mathematics, University of Utah, 155 South 1400 East, SaltLake City, Utah 84112, USAE-mail: [email protected]
Abstract.
Stochastic hybrid systems involve a coupling between a discreteMarkov chain and a continuous stochastic process. If the latter evolvesdeterministically between jumps in the discrete state, then the system reducesto a piecewise deterministic Markov process (PDMP). Well known examplesinclude stochastic gene expression, voltage fluctuations in neurons, and motor-driven intracellular transport. In this paper we use coherent spin states toconstruct a new path integral representation of the probability density functionalfor stochastic hybrid systems, which holds outside the weak noise regime. We usethe path integral to derive a system of Langevin equations in the semi-classicallimit, which extends previous diffusion approximations based on a quasi-steady-state reduction. We then show how in the weak noise limit the path integral isequivalent to an alternative representation that was previously derived using Doi-Peliti operators. The action functional of the latter is related to a large deviationprinciple for stochastic hybrid systems.
Key Words : stochastic hybrid systems, spin coherent states, path integrals, leastaction principles a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b pin coherent states and stochastic hybrid path integrals
1. Introduction
Stochastic hybrid systems, which involve a coupling between a discrete Markov chainand a continuous stochastic process, are finding an increasing number of applicationsin biology [6, 11]. One of the simplest examples is a velocity jump process describinga particle randomly switching between different velocity states. The particle couldrepresent a bacterial cell undergoing chemotaxis [1, 27] or a motor-cargo complexwalking along a cytoskeletal filament [44, 23, 36, 37, 4]. A second example is aconductance-based model of a neuron [21, 14, 31, 24, 13, 39, 7, 40], which considersthe dynamics of the membrane voltage under the stochastic opening and closing ofmembrane-bound ion channels whose transition rates are voltage dependent. Thenumber of open ion channels at time t is represented by a discrete random variable N ( t )and the voltage by a continuous random variable X ( t ). Another important applicationis stochastic gene expression, where N ( t ) could represent the activity state of a gene(due to the binding/unbinding of transcription factors) and X ( t ) the concentrationof a synthesized protein [32, 30, 38, 41, 28]. (Note, however, that in the case of lowprotein concentrations, one has to keep track of the protein number, which is modeledas a second discrete process [45, 50, 2].) A final example is a stochastic hybrid neuralnetwork of synaptically coupled neuronal populations [4, 8, 49]; the state of each localpopulation is described in terms of two stochastic variables, a continuous synapticvariable and a discrete activity variable.A piecewise deterministic Markov process (PDMP) is a particular subsetof stochastic hybrid systems, in which the continuous random variables evolvedeterministically between jumps in the discrete random variables [15]. PDMPs havebeen studied extensively within the context of large deviation theory [33, 19, 20, 10].One major finding of these studies is that the rate function of the associated largedeviation principle (LDP) can be related to an action functional, whose Hamiltoniancorresponds to the principal eigenvalue of a linear operator. The latter incorporatesboth the generator of the discrete Markov process and the vector fields of the piecewisedeterministic dynamics. An alternative method for deriving the action is to constructa path integral representation of the probability density functional in the weak noiselimit. We originally derived such a path integral using integral representations of Diracdelta functions [4, 8], analogous to the analysis of stochastic differential equations(SDEs) [34, 18, 29]. (This approach, which avoids the use of “quantum-mechanical”operators, can also be applied to master equations by considering differential equationsfor the corresponding generator or marginalized distribution of the Markov process[48].) Recently, we developed a more efficient and flexible framework for constructinghybrid path integrals in the weak noise limit [12], which combines the Doi-Pelitioperator formalism for master equations [16, 17, 42, 48] with an analogous operatormethod for SDEs [46].One of the major steps in the derivation of the Doi-Peliti path integral for masterequations is to project the discrete states onto an overdetermined set of coherent“bosonic” states. This is particularly useful when the number of discrete states isunbounded, as in a variety of birth-death processes. However, when the number ofdiscrete states is two or three, say, then a more natural decomposition is in terms ofcoherent “spin” states [43, 22]. Such a decomposition has recently been used to studystochastic gene expression in the presence of promoter noise and low protein copynumbers [45, 50, 2]. For example, one can effectively map the stochastic dynamics ofa single genetic switch to a quantum spin-boson system. In this paper we use coherent pin coherent states and stochastic hybrid path integrals
2. Two-state stochastic hybrid systems
In order to develop the basic theory, consider a stochastic hybrid system whose stateat time t consists of the pair ( X ( t ) , N ( t )), where X ( t ) ∈ R and N ( t ) ∈ { , } . Supposethat the discrete process evolves according to the two-state Markov chain0 β (cid:10) α . (2.1)We also allow the transition rates to depend on the continuous state variable X ( t ),that is, α = α ( x ) , β = β ( x ) for X ( t ) = x . In between jumps in the discrete variable, X ( t ) evolves according to the Ito SDE dX = F n ( x ) dt + (cid:112) D n ( x ) dW (2.2)for N ( t ) = n , where W ( t ) is a Wiener process with (cid:104) W ( t ) (cid:105) = 0 , (cid:104) W ( t ) W ( t (cid:48) ) (cid:105) = min { t, t (cid:48) } . Introduce the probability density P n ( x, t ) dx = Prob { X ( t ) ∈ ( x, x + dx ) , N ( t ) = n } , (2.3)given an initial state X (0) = x , N (0) = n . The probability density evolves accordingto the differential CK equation ∂P n ( x, t ) ∂t = − ∂F n ( x ) P n ( x, t ) ∂x + ∂ D n ( x ) P n ( x, t ) ∂x + (cid:88) m =0 , Q nm ( x ) P m ( x, t ) , (2.4)with matrix generator Q = (cid:18) − β ( x ) α ( x ) β ( x ) − α ( x ) (cid:19) . (2.5)One of the simplest examples of a two-state hybrid system is a gene network withautoregulatory feedback, see Fig. 1(a). Let x ( t ) denote the concentration of protein X at time t and let N ( t ) represent the current state of the gene. If N ( t ) = 0 then thegene is active and synthesizes the protein at a rate κ , whereas if N ( t ) = 1 then thegene is inactive and protein production halts. We thus have the PDMP dxdt = F n ( x ) ≡ κ (1 − n ) − γx, (2.6) pin coherent states and stochastic hybrid path integrals (a) κ γα β Xregulatory feedbackmotor microtubule -v1 v0 βα x (b) Figure 1.
Examples of two-state hybrid systems. (a) An autoregulatory network.A gene is activated (or repressed) by its own protein product X when it binds toa promoter site. (b) Bidirectional motor transport. where γ is the protein degradation rate. Now suppose that the gene is active when oneof its operator sites is bound by X and inactive when it is unbound. Switching betweenthe inactive and active states is then controlled by protein binding/unbinding. Inparticular, we can identify α and β with the binding and unbinding rates, respectively.Moreover, α will depend on the protein concentration x due to the autoregulatoryfeedback. The kinetic equation (2.6) ignores any fluctuations in the number of proteinsdue to finite-size effects. Let the number of proteins at time t be M ( t ) = x ( t )Ω whereΩ is the system-size (cell volume, say). If we include both promoter and proteinfluctuations, then the stochastic dynamics is described by a master equation for thejoint probability distribution P ( n, m, t ) = P [ N ( t ) = n, M ( t ) = m ], where m ≥ n ∈ { , } [45, 50, 28]: dP ( n, m, t ) dt = [ βn + α (1 − n )] P (1 − n, m, t ) + κ (1 − n ) P ( n, m − , t ) (2.7)+ γ ( m + 1) P ( n, m + 1 , t ) − [ β (1 − n ) + αn + κ (1 − n ) + γm ] P ( n, m, t ) , with P ( n, − , t ) = 0. A stochastic hybrid system of the form (2.2) can then beobtained by carrying out a system-size expansion of the master equation with F n ( x ) = κ (1 − n ) − γx, D n ( x ) = Ω − ( κ (1 − n ) + γx ) . (2.8)A second example of a two-state hybrid system is bidirectional motor transportwithin cells [36, 37, 3]. Consider a particle moving along a one-dimensional track,see Fig. 1(b). The particle could represent a motor-cargo complex and the trackcould represent a set of microtubular filaments in the axon of a neuron. ‡ The particle ‡ Microtubules are polarized polymeric filaments with biophysically distinct (+) and ( − ) ends, andthis polarity determines the preferred direction in which an individual molecular motor moves. Forexample, kinesin moves towards the (+) end whereas dynein moves towards the ( − ) end. Onemechanism for bidirectional transport is a tug-of-war model between opposing groups of processivemotors [26, 35]. pin coherent states and stochastic hybrid path integrals v and aleft-moving (retrograde) state with speed v . If N ( t ) denotes the velocity state at time t , then the position x ( t ) of the motor evolves according to the simple PDMP dxdt = (1 − n ) v n (2.9)for N ( t ) = n . One mechanism for generating an x -dependent transition rate involvesmicrotubule associated proteins (MAPs). These molecules bind to microtubules andeffectively modify the free energy landscape of motor-microtubule interactions. Forexample, tau is a MAP found in the axon of neurons and is known to be a key playerin Alzheimer’s disease. Experiments have shown that tau can significantly alter thedynamics of kinesin; specifically, by reducing the rate at which kinesin binds to themicrotubule [47]. Thus tau signaling can be incorporated into a motor transport modelby considering a tau concentration-dependent kinesin binding rate [37]. In terms ofthe simplified two-state PDMP, this means that the rate of switching to the right-moving state becomes a decreasing function of the τ concentration c . Assuming thatthe latter varies with x , we have α ( x ) = α ( c ( x )) , α (cid:48) ( c ) <
0. Finally, one obtains astochastic hybrid system of the form (2.2) if one also takes into account a diffusivecomponent in the motion of the particle.
3. Operator formalism
Recently we combined operator formulations of master equations [16, 17, 42] and SDEs[46] to rewrite the CK equation (2.4) as an operator equation acting on a Hilbert space[12]. This was then used to derive a corresponding hybrid path integral in the weaknoise limit. In order to derive a path integral that holds for arbitrary levels of noise, weconsider an alternative operator formalism that is particularly useful when the numberof discrete states is small. The basic idea is to replace the bosonic annihilation andcreation operators of Doi-Peliti with Pauli spin operators acting on coherent spinstates. The latter have also recently been used to study the effects of promoter noisein gene networks [45, 50, 2]. For the sake of clarity, we introduce the continuous anddiscrete operator constructions separately, and then show to combine them in thecase of the hybrid system. (For ease of notation, we suppress the x -dependence of thetransition rates However, all of the results hold if such x -dependence is included, unlessstated otherwise. We also take x ∈ R although the actual dynamics may restrict thedomain of x . For example, in the case of a gene network x represents a concentrationso that we require x ≥ Consider the following Ito SDE for X ( t ) ∈ R : dX ( t ) = A ( X ) dt + (cid:112) D ( X ) dW ( t ) . (3.1)The corresponding Fokker-Planck (FP) equation for the probability density P ( x, t ) is ∂P ( x, t ) ∂t = − ∂A ( x ) P ( x, t ) ∂x + ∂ D ( x ) P ( x, t ) ∂x . (3.2)Following Ref. [46, 12], we introduce a Hilbert space spanned by the vectors | x (cid:105) ,together with a conjugate pair of position-momentum operators ˆ x and ˆ p such that[ˆ x, ˆ p ] = i. (3.3 a ) pin coherent states and stochastic hybrid path integrals x | x (cid:105) = x | x (cid:105) , ˆ p | x (cid:105) = − i ← ddx | x (cid:105) . (3.3 b )The arrow on the differential operator indicates that it operates to the left.Alternatively, given a state vector | φ (cid:105) = (cid:82) ∞−∞ dxφ ( x ) | x (cid:105) , we have (cid:104) x | ˆ p | φ (cid:105) = − iφ (cid:48) ( x ).The inner product and completion relations on the Hilbert space are (cid:104) x (cid:48) | x (cid:105) = δ ( x − x (cid:48) ) , (cid:90) ∞−∞ dx | x (cid:105)(cid:104) x | = 1 . (3.4)Given the probability density P ( x, t ) we define the state vector | ψ ( t ) (cid:105) = (cid:90) ∞−∞ dx P ( x, t ) | x (cid:105) . (3.5)Differentiating both sides with respect to time t and using the FP equation gives ddt | ψ ( t ) (cid:105) = (cid:90) ∞−∞ dx (cid:20) − ∂A ( x ) P ( x, t ) ∂x + ∂ D ( x ) P ( x, t ) ∂x (cid:21) | x (cid:105) = (cid:90) ∞−∞ dx − A ( x ) P ( x, t ) ← ∂∂x + D ( x ) P ( x, t ) ← ∂ ∂x | x (cid:105) = (cid:2) − i ˆ pA (ˆ x ) − ˆ p D (ˆ x ) (cid:3) (cid:90) ∞−∞ dx P ( x, t ) | x (cid:105) . Hence, we can write the FP equation in the operator form ddt | ψ ( t ) (cid:105) = ˆ H fp | ψ ( t ) (cid:105) , (3.6 a )with ˆ H fp = − i ˆ pA (ˆ x ) − ˆ p D (ˆ x ) . (3.6 b )The formal solution of the FP equation is | ψ ( t ) (cid:105) = e ˆ H fp t | ψ (0) (cid:105) , (3.7)and expectations are given by (cid:104) X ( t ) (cid:105) = (cid:90) ∞∞ dx xP ( x, t ) = (cid:90) ∞−∞ dx (cid:104) x | ˆ x | ψ ( t ) (cid:105) = (cid:90) ∞−∞ dx (cid:104) x | ˆ x e ˆ H fp t | ψ (0) (cid:105) . (3.8)Another useful choice of basis vectors is the momentum representation (analogousto taking Fourier transforms), | p (cid:105) = (cid:90) ∞−∞ dx e ipx | x (cid:105) . (3.9)It immediately follows that | p (cid:105) is an eigenvector of the momentum operator ˆ p , sinceˆ p | p (cid:105) = (cid:90) ∞−∞ dx e ipx − i ← ddx | x (cid:105) = (cid:90) ∞−∞ dx p e ipx | x (cid:105) = p | p (cid:105) . (3.10)Using the inverse Fourier transform, we also have | x (cid:105) = (cid:90) ∞−∞ dp π e − ipx | p (cid:105) , (3.11)and the completeness relation (cid:90) ∞−∞ dp π | p (cid:105)(cid:104) p | = 1 . (3.12) pin coherent states and stochastic hybrid path integrals Consider the master equation for a two-state Markov chain, written in matrix form d P dt = QP ( t ) , P ( t ) = ( P ( t ) , P ( t )) (cid:62) , (3.13)with Q the matrix (2.5). Introduce the Pauli spin matrices σ x = 12 (cid:18) (cid:19) , σ y = 12 (cid:18) − ii (cid:19) , σ z = 12 (cid:18) − (cid:19) , (3.14)and set σ ± = σ x ± iσ y . (3.15)It follows that the generator can be rewritten as Q = − β (cid:18) + σ z (cid:19) − α (cid:18) − σ z (cid:19) + ασ + + βσ − . (3.16)Next we define the coherent spin-1 / | s (cid:105) = (cid:18) e iφ/ cos θ/ − iφ/ sin θ/ (cid:19) , ≤ θ ≤ π, ≤ φ < π, (3.17)together with the adjoint (cid:104) s | = (cid:16) e − iφ/ , e iφ/ (cid:17) . (3.18)Note that (cid:104) s (cid:48) | s (cid:105) = e i ( φ − φ (cid:48) ) / cos θ/ − i ( φ − φ (cid:48) ) / sin θ/ , (3.19)so that (cid:104) s | s (cid:105) = 1 and (cid:104) s + ∆ s | s (cid:105) = 1 − i ∆ φ cos θ + O (∆ φ ) . (3.20)We also have the completeness relation12 π (cid:90) π sin θ dθ (cid:90) π dφ | s (cid:105)(cid:104) s | = 1 . (3.21)It can checked that the following identities hold: (cid:104) s | σ z | s (cid:105) = 12 cos θ, (3.22 a ) (cid:104) s | σ + | s (cid:105) = 12 e iφ sin θ, (3.22 b ) (cid:104) s | σ − | s (cid:105) = 12 e − iφ sin θ. (3.22 c )Hence, (cid:104) s | Q | s (cid:105) = Q ( θ, φ ) ≡ − β (cid:0) − e iφ (cid:1) θ − α (cid:0) − e − iφ (cid:1) − cos θ . (3.23) pin coherent states and stochastic hybrid path integrals Let us now return to the CK equation (2.4). Introduce the state vectors | ψ n ( t ) (cid:105) = (cid:90) ∞−∞ dx P n ( x, t ) | x (cid:105) , n = 0 , , (3.24)and rewrite (2.4) in the operator form ddt | ψ n ( t ) (cid:105) = (cid:2) − i ˆ pF n (ˆ x ) − ˆ p D n (ˆ x ) (cid:3) | ψ n ( t ) (cid:105) + (cid:88) m =0 , Q nm | ψ m ( t ) (cid:105) . Set (cid:98) H n = − i ˆ pF n (ˆ x ) − ˆ p D n (ˆ x ) , n = 0 , , (3.25)and consider the diagonal matrix operator (cid:98) K = (cid:32) (cid:98) H (cid:98) H (cid:33) = (cid:18) + σ z (cid:19) (cid:98) H + (cid:18) − σ z (cid:19) (cid:98) H . (3.26)We can thus rewrite (2.4) as an operator equation ddt | ψ ( t ) (cid:105) = (cid:98) H | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) = ( | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) ) (cid:62) , (3.27)with (cid:98) H = (cid:98) K + Q (3.28)= (cid:18) + σ z (cid:19) (cid:98) H + (cid:18) − σ z (cid:19) (cid:98) H − β (cid:18) + σ z (cid:19) − α (cid:18) − σ z (cid:19) + ασ + + βσ − . Moreover, (cid:104) s | (cid:98) H | s (cid:105) = H ( θ, φ, ˆ x, ˆ p ) ≡ − (cid:16) β (cid:2) − e iφ (cid:3) − (cid:98) H (cid:17) θ − (cid:16) α (cid:2) − e − iφ (cid:3) − (cid:98) H (cid:17) − cos θ . (3.29)Formally integrating equation (3.27) gives | ψ ( t ) (cid:105) = e (cid:98) H t | ψ (0) (cid:105) . (3.30) The above construction can be extended to three or more states, although the analysisbecomes more complicated. Here we will restrict our discussion to a three-state modelwith N ( t ) ∈ { , , } and a matrix generator of the form Q = − β + α + β + − α + − α − β − α − − β − . (3.31)This discrete process has recently appeared in a stochastic model of gene expressionthat includes three distinct histone states as well as two DNA promoter states [2] § .Another example is the three-state model of motor transport shown in Fig. 2, whichconsists of a right-moving state ( n = 0 , v = v ), a stationary state ( n = 1 , v = 0),and a left-moving state ( n = 2 , v = − v ). Moreover, transitions can only occur eitherinto or out of the stationary state. § Histones are proteins found in eukaryotic cell nuclei that pack and order the DNA into structuralunits called nucleosomes. They play an important role in epigenetics. pin coherent states and stochastic hybrid path integrals -v v β - β + α - α + motor microtubule x Figure 2.
Three-state motor transport model.
We develop the coherent spin-1 state construction along the lines of [2]. First,rewrite the generator (3.31) in the form Q = α + T + + α − T − + β + S + + β − S − , (3.32)with T + = − , T − = − , (3.33)and S + = − , S − = − , (3.34)Next we introduce the coherent spin-1 state | s (cid:105) = e iφ cos θ/
22 cos θ/ θ/ − iφ sin θ/ , ≤ θ ≤ π, ≤ φ < π, (3.35)together with the adjoint (cid:104) s | = (cid:0) e − iφ , , e iφ (cid:1) . (3.36)Note that (cid:104) s (cid:48) | s (cid:105) = e i ( φ − φ (cid:48) ) cos θ + 2 cos θ/ θ/ − i ( φ − φ (cid:48) ) sin θ/ , (3.37)so that (cid:104) s | s (cid:105) = 1 and (cid:104) s + ∆ s | s (cid:105) = 1 − i ∆ φ (cos θ/ − sin θ/
2) + O (∆ φ ) ≈ − i ∆ φ cos θ. (3.38)We also have the completeness relation34 π (cid:90) π sin θ dθ (cid:90) π dφ | s (cid:105)(cid:104) s | = 1 . (3.39)It can checked that the following identities hold: (cid:104) s | T + | s (cid:105) = 2(e − iφ −
1) cos θ/ θ/ , (3.40 a ) (cid:104) s | T − | s (cid:105) = 2(e iφ −
1) cos θ/ θ/ , (3.40 b ) (cid:104) s | S + | s (cid:105) = (e iφ −
1) cos θ/ , (3.40 c ) (cid:104) s | S − | s (cid:105) = (e − iφ −
1) sin θ/ . (3.40 d )Hence, (cid:104) s | Q | s (cid:105) = Q ( θ, φ ) ≡ − (cid:0) − e iφ (cid:1) (cid:0) α − cos θ/ θ/ β + cos θ/ (cid:1) (3.41) − (cid:0) − e − iφ (cid:1) (cid:0) α + cos θ/ θ/ β − sin θ/ (cid:1) . pin coherent states and stochastic hybrid path integrals ddt | ψ ( t ) (cid:105) = (cid:98) H | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) = ( | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) ) (cid:62) , (3.42)with (cid:104) s | (cid:98) H | s (cid:105) = Q ( θ, φ ) + (cid:98) H cos θ/ (cid:98) H cos θ/ θ/ (cid:98) H sin θ/ , (3.43)and (cid:98) H n = − i ˆ pF n (ˆ x ) − ˆ p D n (ˆ x ) , n = 0 , , . (3.44)
4. Construction of stochastic hybrid path integral
One of the advantages of expressing the evolution equation for the probability densityin terms of an operator equation acting on a Hilbert space is that it is relativelystraightforward to construct a corresponding path integral representation of thesolution. For simplicity, we consider the two-state model and then indicate how toextend the construction to the three-state model. As with other stochastic processes,the first step is to divide the time interval [0 , t ] into N subintervals of size ∆ t = t/N and rewrite the formal solution (3.30) as | ψ ( t ) (cid:105) = e (cid:98) H ∆ t e (cid:98) H ∆ t · · · e (cid:98) H ∆ t | ψ (0) (cid:105) , (4.1)with (cid:98) H given by equation (3.28). We then insert multiple copies of appropriatelychosen completeness relations. Here we use the completeness relations (3.4) and (3.21),which are applied to the product Hilbert space with | s, x (cid:105) = | s (cid:105) ⊗ | x (cid:105) . Introducing thesolid angle integral (cid:90) Ω ds = 12 π (cid:90) π sin θ dθ (cid:90) π dφ, (4.2)we have | ψ ( t ) (cid:105) = (cid:90) Ω ds · · · (cid:90) Ω ds N (cid:90) ∞−∞ dx · · · (cid:90) ∞−∞ dx N | s N , x N (cid:105)× (cid:104) s N , x N | e (cid:98) H ∆ t | s N − , x N − (cid:105)(cid:104) s N − , x N − | e (cid:98) H ∆ t | s N − , x N − (cid:105)· · · × (cid:104) s , x | e (cid:98) H ∆ t | s , x (cid:105)(cid:104) s , x | ψ (0) (cid:105) . (4.3)In the limit N → ∞ and ∆ t → N ∆ t = t fixed, we can make theapproximation (cid:104) s j +1 , x j +1 | e (cid:98) H ∆ t | s j , x j (cid:105) ≈ (cid:104) s j +1 , x j +1 | (cid:98) H ∆ t | s j , x j (cid:105) = (cid:104) s j +1 | s j (cid:105) (cid:26) δ ( x j +1 − x j ) + (cid:104) x j +1 | H ( θ j , φ j , x j , ˆ p j )∆ t | x j (cid:105) (cid:27) + O (∆ t ) , (4.4)with H defined in equation (3.29). In addition, equation (3.20) implies that (cid:104) s j +1 | s j (cid:105) = 1 − i ( φ j +1 − φ j ) cos θ j + O (∆ φ ) = 1 − i ∆ t dφ j dt cos θ j + O (∆ t ) . (4.5)Each small-time propagator thus becomes (to first order in ∆ t ) (cid:104) s j +1 , x j +1 | e (cid:98) H ∆ t | s j , x j (cid:105) (4.6) ≈ (cid:104) x j +1 | exp (cid:18)(cid:20) H ( θ j , φ j , x j , ˆ p j ) − i dφ j dt cos θ j (cid:21) ∆ t (cid:19) | x j (cid:105) . pin coherent states and stochastic hybrid path integrals (cid:104) s j +1 , x j +1 | e (cid:98) H ∆ t | s j , x j (cid:105)≈ (cid:90) ∞−∞ dp j π (cid:104) x j +1 | p j (cid:105)(cid:104) p j | x j (cid:105) exp (cid:18)(cid:20) H ( θ j , φ j , x j , p j ) − i dφ j dt cos θ j (cid:21) ∆ t (cid:19) . (4.7)Furthermore, (cid:104) x j +1 | p j (cid:105)(cid:104) p j | x j (cid:105) = e ip j ( x j +1 − x j ) = exp (cid:18) ip j dx j dt ∆ t (cid:19) + O (∆ t ) . (4.8)Substituting equations (4.7) and (4.8) into (4.3) yields | ψ ( t ) (cid:105) = (cid:90) Ω ds · · · (cid:90) Ω ds N (cid:90) ∞−∞ (cid:90) ∞−∞ dx dp π · · · (cid:90) ∞−∞ (cid:90) ∞−∞ dx N dp N π | s N , x N (cid:105)× N − (cid:89) j =0 exp (cid:18)(cid:20) H ( θ j , φ j , x j , p j ) − i dφ j dt cos θ j + ip j dx j dt (cid:21) ∆ t (cid:19) (cid:104) s , x | ψ (0) (cid:105) . The final step is to take the continuum limit N → ∞ , ∆ t → N ∆ t = t fixed, x j = x ( j ∆ t ) etc. We will also assume that < x | ψ n (0) (cid:105) = ρ n δ ( x − x ) , and set P n ( x, t | x ,
0) = (cid:104) x, n | ψ ( t ) (cid:105) . After Wick ordering, p → − ip , integrating by parts the term involving dφ/dt , andperforming the change of coordinates z = (1 + cos θ ) /
2, we obtain the followingfunctional path integral: P n ( x, t | x ,
0) = N n (cid:90) x ( t )= xx (0)= x D [ φ ] D [ z ] D [ p ] D [ x ] exp (cid:18) − (cid:90) t (cid:20) p dxdτ − iφ dzdτ − H (cid:21) dτ (cid:19) , (4.9)where N n is a constant and H is the effective “Hamiltonian” H = (cid:0) − β (cid:2) − e iφ (cid:3) + pF ( x ) + p D ( x ) (cid:1) z + (cid:0) − α (cid:2) − e − iφ (cid:3) + pF ( x ) + p D ( x ) (cid:1) (1 − z ) , (4.10)with “position” coordinates ( x, z ) and “conjugate” momenta ( p, − iφ ). It will turn outthat z ( t ) represents the probability that the discrete state N ( t ) = 0 at time t . (i) If F and D are independent of the discrete state n and the transition rates α, β are x -independent, then the path integral (4.9) reduces to the product of two independentpath integrals, corresponding to the continuous and discrete processes, respectively: P n ( x, t | x ,
0) = x ( t )= x (cid:90) x (0)= x D [ p ] D [ x ] exp (cid:18) − (cid:90) t [ p ˙ x − pF ( x ) − p D ( x )] dτ (cid:19) (4.11) × N n (cid:90) D [ φ ] D [ z ] exp (cid:18) − (cid:90) t (cid:2) − iφ ˙ z + β (cid:2) iφ (cid:3) z + α (cid:2) − iφ (cid:3) (1 − z ) (cid:3) dτ (cid:19) . pin coherent states and stochastic hybrid path integrals S [ x, p ] = (cid:90) t [ p ˙ x − pF ( x ) − p D ( x )] dτ. (4.12)(iii) The derivation carries over to higher-dimensional stochastic hybrid systems with M continuous variables x (cid:96) , (cid:96) = 1 , . . . M . The Ito SDE becomes dX (cid:96) = F n,(cid:96) ( x ) dt + (cid:113) D n,(cid:96) ( x ) dW (cid:96) (4.13)for N ( t ) = n , where W (cid:96) ( t ) are independent Wiener processes. The multivariate CKequation takes the form ∂P n ∂t = M (cid:88) (cid:96) =1 (cid:20) − ∂∂x (cid:96) ( F n,(cid:96) ( x ) P n ( x , t )) + ∂ ∂x (cid:96) ( D n,(cid:96) ( x ) P n ( x , t )) (cid:21) + (cid:88) m Q nm P m ( x , t ) . (4.14)Following along identical lines to the one-dimensional case, one obtains a path-integralrepresentation of the solution to equation (4.14): P n ( x , t | x ,
0) = N n (cid:90) x ( t )= xx (0)= x D [ φ ] D [ z ] D [ p ] D [ x ] × exp (cid:32) − (cid:90) t (cid:34) M (cid:88) (cid:96) =1 p (cid:96) ˙ x (cid:96) − iφ ˙ z − H (cid:35) dτ (cid:33) , (4.15)with H = (cid:32) − β (cid:2) − e iφ (cid:3) + M (cid:88) (cid:96) =1 [ p (cid:96) F ,(cid:96) ( x ) + p (cid:96) D ,(cid:96) ( x )] (cid:33) z + (cid:32) − α (cid:2) − e − iφ (cid:3) + M (cid:88) (cid:96) =1 [ p (cid:96) F ,(cid:96) ( x ) + p (cid:96) D ,(cid:96) ( x )] (cid:33) (1 − z ) . (4.16)(iii) The path integral construction can also be extended to the case of more than twodiscrete states. In particular, consider the three-state model of section 3.4. All of thesteps in the derivation proceed as before. The final result is a path integral of theform (4.9) with dz/dt → dz/dt and the modified Hamiltonian H = (cid:0) − β + (cid:2) − e iφ (cid:3) + pF ( x ) + p D ( x ) (cid:1) z (4.17)+ 2 (cid:0) − α − (cid:2) − e iφ (cid:3) − α + (cid:2) − e − iφ (cid:3) + pF ( x ) + p D ( x ) (cid:1) z (1 − z )+ (cid:0) − β − (cid:2) − e − iφ (cid:3) + pF ( x ) + p D ( x ) (cid:1) (1 − z ) . In general it is not possible to evaluate a stochastic path integral without someform of approximation scheme. One of the best known is the so-called semi-classicalapproximation, which involves expanding the path integral action to second order inthe variables p, φ , assuming that the system operates in the weak noise regime. Inthe case of the piecewise SDE (2.2), we define the weak noise limit by introducing thescalings α → α/(cid:15), β → β/(cid:15) and D → (cid:15)D . The former represents fast switching between pin coherent states and stochastic hybrid path integrals φ → (cid:15)φ , the path integral (4.9) for the two-statemodel becomes P n ( x, t | x ,
0) = N n (cid:90) x ( t )= xx (0)= x D [ φ ] D [ z ] D [ p ] D [ x ]e − S , (4.18)with the action S = (cid:90) t (cid:20) p ˙ x − i(cid:15)φ ˙ z + (cid:18) β(cid:15) (cid:2) − e i(cid:15)φ (cid:3) − pF ( x ) − (cid:15)p D ( x ) (cid:19) z + (cid:16) α(cid:15) (cid:2) − e − i(cid:15)φ (cid:3) − pF ( x ) − (cid:15)p D ( x ) (cid:17) (1 − z ) (cid:21) dτ. (4.19)Under the approximation 1 − e ± i(cid:15)φ = ∓ i(cid:15)φ + (cid:15) φ / . . . , the action is quadratic in p, φ . Comparison with the action (4.12) of an SDE then establishes that the resultingpath integral represents the probability density functional of an effective stochasticprocesses evolving according to the following pair of coupled Langevin equations: dxdt = F ( x ) z + F ( x )(1 − z ) + √ (cid:15)ξ x , (4.20 a ) (cid:15) dzdt = − βz + α (1 − z ) + √ (cid:15)ξ z , (4.20 b )where ξ x and ξ z are independent Gaussian white noise processes with (cid:104) ξ x (cid:105) = 0 = (cid:104) ξ z (cid:105) and (cid:104) ξ x ( t ) ξ x ( t (cid:48) ) (cid:105) = [ D ( x ) z + D ( x )(1 − z )] δ ( t − t (cid:48) ) , (4.21 a ) (cid:104) ξ z ( t ) ξ z ( t (cid:48) ) (cid:105) = 12 ( βz + α (1 − z )) δ ( t − t (cid:48) ) . (4.21 b )Note that the multiplicative noise term ξ z in equation (4.20 b ) vanishes at z = 0 , z ( t ) remains within the domain [0 , z ( t ) as an auxiliary variable that represents the effective probabilitythat N ( t ) = 0 at time t . An analogous result holds for the 3-state model of section3.4. That is, z ( t ) parameterizes an effective probability distribution ψ n , n = 0 , , ψ = z , ψ = 2 z (1 − z ) and ψ = (1 − z ) . (More generally, for N discrete states ψ n is generated by considering the binomial expansion of ( z + (1 − z )) N .)A further approximation can be obtained by using a linear noise approximation.Taking the limit (cid:15) → z ( t ) → z ∗ = α/ ( α + β ) and x ( t ) satisfies thedeterministic mean-field equation dxdt = F ( x ) = F ( x ) z ∗ ( x ) + F ( x )(1 − z ∗ ( x )) , (4.22)assuming x -dependent transition rates. Clearly ρ = z ∗ and ρ = 1 − z ∗ is thestationary distribution of the two-state Markov chain (3.13). Substituting z ( t ) = z ∗ + y ( t ) into equation (4.21 b ) implies that to leading order, y ( t ) = √ (cid:15)α + β ξ ( t ) , with (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = βz ∗ δ ( t − t (cid:48) ) . Applying the linear noise approximation to equation (4.21 a ) then gives dxdt = F ( x ) + [ F ( x ) − F ( x )] y + √ (cid:15)ξ x = F ( x ) + √ (cid:15)ξ + √ (cid:15)ξ x , (4.23) pin coherent states and stochastic hybrid path integrals (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = D eff ( x ) δ ( t − t (cid:48) ) , D eff ( x ) ≡ αβα + β [ F ( x ) − F ( x )] ( α + β ) . (4.24)A little algebra shows that D eff ( x ) = α [ F ( x ) − F ( x )] + β [ F ( x ) − F ( x )] ( α + β ) , which is precisely the effective diffusion coefficient obtained using a quasi-steady-stateapproximation of the CK equation ∂P n ( x, t ) ∂t = − ∂F n ( x ) P n ( x, t ) ∂x + 1 (cid:15) (cid:88) m =0 , Q nm ( x ) P m ( x, t ) , (4.25)in the fast switching limit [36]. The latter method is based on substituting the solution P n ( x, t ) = C ( x, t ) ρ n + (cid:15)w n ( x, t ) into the CK equation and deriving a Fokker-Planckequation for C ( x, t ) using the Liapunov-Schmidt procedure. It essentially assumesthat z ( t ) ≈ z ∗ as in the linear noise approximation. One of the nice features of thepath integral representation based on coherent spin states is that it also keeps trackof z ( t ). In particular, one can explore the stochastic dynamics under the semiclassicalapproximation in the weakly nonadiabatic regime, by considering the coupled systemof Langevin equations (4.21 a ) and (4.21 b ). This observation has also been made withinthe specific context of stochastic gene expression [50]. As a simple illustration of the semi-classical limit, consider a two-state gene networkwithout feedback, see equation (2.6) and Fig. 2. (In the limit γ → v = κ and v = 0.) The pair ofLangevin equations takes the form (after setting z = z ∗ + y ) dxdt = κz ∗ − γx + κy, (cid:15) dydt = − ( α + β ) y + √ (cid:15)ξ y , (4.26)with α, β constant, z ∗ = α/ ( α + β ) and (cid:104) ξ y ( t ) ξ y ( t (cid:48) ) (cid:105) = 12 (2 βz ∗ + ( β − α ) y ) δ ( t − t (cid:48) ) . (4.27)Integrating the equations (4.26) with respect to t yields x ( t ) = x e − γt + κz ∗ γ (cid:0) − e − γt (cid:1) + κ (cid:90) t e − γ ( t − t (cid:48) ) y ( t (cid:48) ) dt (cid:48) , (4.28)and y ( t ) = y e − ( α + β ) t/(cid:15) + (cid:114) (cid:15) (cid:90) t e − ( α + β )( t − t (cid:48) ) /(cid:15) ξ y ( t (cid:48) ) dt (cid:48) . (4.29)Taking expectations of these two equations and substituting for ¯ y ( t ) into (4.28), weobtain the following equation for the mean protein concentration ¯ x :¯ x ( t ) = x e − γt + κz ∗ γ (cid:0) − e − γt (cid:1) + (cid:15)κy ( α + β ) − (cid:15)γ (cid:16) e − γt − e − ( α + β ) t/(cid:15) (cid:17) . (4.30)It follows that taking into account the dynamics of the auxiliary variable z ( t ) leads toadditional contributions to the dynamics of the mean protein concentration that are pin coherent states and stochastic hybrid path integrals (cid:104) [ x ( T ) − ¯ x ( T )] (cid:105) = 2 κ (cid:90) T (cid:90) t e − γ ( T − t ) e − γ ( T − t (cid:48) ) ∆ y ( t, t (cid:48) ) dt (cid:48) dt, (4.31)where (for t (cid:48) < t )∆ y ( t, t (cid:48) ) ≡ (cid:104) [ y ( t ) − ¯ y ( t )][ y ( t (cid:48) ) − ¯ y ( t (cid:48) )] (cid:105) = 2 (cid:15) (cid:90) t e − Γ( t − τ ) (cid:90) t (cid:48) e − Γ( t (cid:48) − τ (cid:48) ) (cid:104) ξ y ( τ ) ξ y ( τ (cid:48) ) (cid:105) dτ (cid:48) dτ = 1 (cid:15) (cid:90) t (cid:48) e − Γ( t + t (cid:48) − τ ) (2 βz ∗ + ( β − α ))¯ y ( τ ) dτ = βz ∗ α + β (cid:104) e − Γ( t − t (cid:48) ) − e − Γ( t + t (cid:48) ) (cid:105) + ( β − α ) y ( α + β ) (cid:16) e − Γ t − e − Γ( t + t (cid:48) ) (cid:17) , (4.32)After some algebra we find that (cid:104) [ x ( T ) − ¯ x ( T )] (cid:105) = 2 κ e − γT α + β (cid:26) βz ∗ γ + Γ (cid:20) e γT − γ − e ( γ − Γ) T − γ − Γ (cid:21) − βz ∗ γ − Γ (cid:20) e γ − Γ) T − γ − Γ) − e ( γ − Γ) T − γ − Γ (cid:21) + ( β − α ) y γ (cid:20) e (2 γ − Γ) T − γ − Γ − e ( γ − Γ) T − γ − Γ (cid:21) − ( β − α ) y γ − Γ (cid:20) e γ − Γ) T − γ − Γ) − e ( γ − Γ) T − γ − Γ (cid:21) (cid:27) . (4.33)In the limit (cid:15) →
0, we can drop all exponentially small terms e − Γ T = e − ( α + β ) T/(cid:15) . Theremaining terms generate a power series in (cid:15) whose leading order form is (cid:104) [ x ( T ) − ¯ x ( T )] (cid:105) = 2 κ α + β (cid:15)βz ∗ α + β − e − γT γ + O ( (cid:15) ) . The O ( (cid:15) ) terms is identical to the variance obtained from the linear noiseapproximation of equations (4.23) and (4.24), which become dxdt = κz ∗ − γx + √ (cid:15)ξ, (4.34)with (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = Dδ ( t − t (cid:48) ) , D ≡ αβα + β κ ( α + β ) . (4.35)(Note that certain care has to be taken in interpreting equation (4.34), since there isa small but non-zero probability that the concentration x ( t ) can become negative.)
5. Alternative path integral representation and least action paths
The form of the path integral (4.9) differs significantly from a previous version thatwas derived in the weak noise limit using either integral representations of the Diracdelta function [4, 8] or operator methods adapted from Doi-Peliti [12]. One of themajor applications of the second version is that it provides a relatively straightforwardmethod for calculating least-action paths in noise-induced escape problems [6].For the moment, let us consider the hybrid system (2.2) with an arbitrary numberof discrete states n = 0 , , . . . , N evolving according to an irreducible Markov chain pin coherent states and stochastic hybrid path integrals Q . The discrete process is said to be irreducible if there exists a t > Q t >
0; this implies that any two states of the Markov chain can beconnected in a finite time. One can then apply the Perron-Frobenius theorem for finitesquare matrices [25]. In particular, there exists a unique positive right-eigenvector ρ n for which (cid:80) m Q nm ρ m = 0; the corresponding left eigenvector is (1 , , . . . ,
1) since (cid:80) n Q nm = 0. We can identify ρ as the unique stationary density. Moreover, thePerron Frobenius theorem ensures that all other eigenvalues have negative real parts,ensuring that the distribution P m ( t ) → ρ m as t → ∞ . Now consider the generalizedeigenvalue equations N (cid:88) m =1 (cid:8) [ pF n ( x ) + p D n ( x )] δ m,n + Q nm ( x ) (cid:9) R m ( x, p ) = Λ( x, p ) R n ( x, p ) , (5.1 a ) (cid:88) m ≥ R m ( x, p ) (cid:8) [ pF n ( x ) + p D n ( x )] δ m,n + Q mn ( x ) (cid:9) = Λ( x, p ) R n ( x, p ) , (5.1 b )with N (cid:88) m =1 R m ( x, p ) R m ( x, p ) = 1 . (5.2)Note that when p = 0 we recover the eigenvalue equation for Q . The Perron-Frobeniustheorem can also be applied to the linear system (5.1 a ), which means that thereexists a unique principal eigenvalue Λ( x, p ) and associated positive eigenvector R ( x, p ).In the weak noise limit (as defined above), one obtains the following path integralrepresentation of the solution to the corresponding CK equation [4, 8, 12]: P n ( x, t ) = N n x ( t )= x (cid:90) x (0)= x D [ p ] D [ x ] exp (cid:18) − (cid:15) (cid:90) t [ p ˙ x − Λ( x, p )] dτ (cid:19) . (5.3)The principal eigenvalue Λ acts as an effective Hamiltonian with the parameter p of theeigenvalue equation (5.1 a ) playing the role of a momentum variable. (The resultingclassical action can also be derived without the use of path integrals [10] using thevariational LDP for hybrid systems introduced by Faggionato et al. [19, 20]; a rigorousbut rather technical derivation can be found in [33].)In the limit (cid:15) →
0, the path integral is dominated by least-action paths, whichsatisfy Hamilton’s equations˙ x = ∂ Λ( x, p ) ∂p , ˙ p = − ∂ Λ( x, p ) ∂x , (5.4)Differentiating both sides of equation (5.1 a ) with respect to p gives N (cid:88) m =1 { [ F n + 2 pD n ] δ m,n + Q nm } R m + N (cid:88) m =1 (cid:8) [ pF n + p D n ] δ m,n + Q nm (cid:9) ∂R m ∂p = Λ ∂R n ∂p + ∂ Λ ∂p R n . Summing over n with (cid:80) n Q nm = 0 and setting p = 0 thus shows that ∂ Λ ∂p (cid:12)(cid:12)(cid:12)(cid:12) p =0 = (cid:88) n F n ρ n = F , pin coherent states and stochastic hybrid path integrals x = F ( x ). (It can also be checkedthat ˙ p = 0 at p = 0.) However, there also exist least action paths for which p (cid:54) = 0.In particular, the zero energy paths with Λ( x, p ) = 0 represent the most likely pathsof escape from a metastable state x ∗ . Evaluating the action along such a path yieldsthe so-called quasipotentialΦ ( x ) = (cid:90) T −∞ p ( t ) ˙ x ( t ) dt, (5.5)with x ( T ) = x and x ( −∞ ) = x ∗ . The latter is also the solution of the Hamilton-Jacobiequation λ ( x, Φ (cid:48) ( x )) = 0 . (5.6)Combining the evaluation of least-action paths with matched asymptotic methodsprovides an estimate for the mean first passage time to escape from a metastablestate, which has the exponential form τ ∼ e Φ ( x ) /(cid:15) [6].In the case of the two-state hybrid model (2.2) it is possible to determine Λexplicitly. The linear equation (5.1 a ) can be written as the two-dimensional system (cid:18) − β + pF ( x ) αβ − α + pF ( x ) (cid:19) (cid:18) R R (cid:19) = Λ (cid:18) R R (cid:19) . (5.7)Solving the corresponding characteristic equation yields the principle eigenvalueΛ( x, p ) = 12 (cid:104) Σ( x, p ) + (cid:112) Σ( x, p ) − x, p ) (cid:105) , (5.8)where Σ( x, p ) = p ( F ( x ) + F ( x )) − [ α + β ] , and Γ( x, p ) = ( pF ( x ) − α )( pF ( x ) − β ) − αβ. A little algebra shows that A ( x, p ) ≡ Σ( x, p ) − x, p ) = [ p ( F − F ) − ( α + β )] + αβ > , so that as expected Λ is real. From Hamilton’s equations˙ x = ∂ Λ( x, p ) ∂p = F ( x ) + F ( x )2 + ∂ A ( x, p ) ∂p (cid:112) A ( x, p )= F ( x ) + F ( x )2 + F ( x ) − F ( x )2 p ( F − F ) − ( α − β ) (cid:112) [ p ( F − F ) + ( α − β )] + αβ . (5.9)Moreover, writing˙ x = F ( x ) ψ ( x, p ) + F ( x ) ψ ( x, p ) , we see that ψ ( x, p ) = 12 (cid:34) p ( F − F ) + ( α − β ) (cid:112) [ p ( F − F ) − ( α − β ))] + αβ (cid:35) , (5.10)and ψ ( x, p ) = 12 (cid:34) − p ( F − F ) + ( α − β ) (cid:112) [ p ( F − F ) − ( α − β )] + αβ (cid:35) , (5.11)so that ψ , ≥ ψ + ψ = 1. This suggests that ψ plays an analogous role tothe dynamical variable z ( t ) in the path integral (4.9). pin coherent states and stochastic hybrid path integrals
6. Equivalence of path integral representations in the weak noise limit
In this section we show how the path integral representation (4.9) reduces to the pathintegral (5.3) in the weak noise limit. At first sight it is not clear how the principaleigenvalue Λ( x, p ) emerges from (4.9) for (cid:15) →
0. Indeed, one has to go beyond the semi-classical limit considered in section 4. We begin by considering the two state modelwith Hamiltonian (4.10). As a first step, consider the scalings α → α/(cid:15), β → β/(cid:15) , D → (cid:15)D and p → p/(cid:15) and rewrite the path integral (4.9) as P n ( x, t | x ,
0) = N n (cid:90) x ( t )= xx (0)= x D [ φ ] D [ z ] D [ p ] D [ x ]e − S/(cid:15) , (6.1)with the action S = (cid:90) t (cid:20) p ˙ x − i(cid:15)φ ˙ z + β (cid:2) − e iφ (cid:3) z + α (cid:2) − e − iφ (cid:3) (1 − z ) − h ( x, p, z ) (cid:21) dτ, (6.2)and h ( x, p, z ) = ( pF ( x ) + p D ( x )) z + ( pF ( x ) + p D ( x ))(1 − z ) . (6.3)(In contrast to equation (4.19), we have not rescaled φ .) Next we define the functions q = β + (cid:2) − e iφ (cid:3) + λ, λ = − α (cid:2) − e − iφ (cid:3) . (6.4)and rewrite the action as S = (cid:90) t (cid:20) p ˙ x − i(cid:15)φ ˙ z + q ( φ ) z − λ ( φ ) − h ( x, p, z ) (cid:21) dτ. (6.5)Note that λ and q are related according to q − β + αβλ + α − λ = 0 . (6.6)Moreover, equation (6.6) is the characteristic equation for the eigenvalue equation( q − β ) r + αr = λr , βr − αr = λr . (6.7)In the limit (cid:15) → (cid:15) dzdt = dqdφ z − dλdφ . (6.8)This allows us to set dz/dt = 0 in the action (6.2) and eliminate the independentvariable φ by requiring dqdφ z − dλdφ = 0 , (6.9)that is, e iφ = α (1 − z ) βz . (6.10)We thus obtain the reduced path integral P n ( x, t | x ,
0) = N n (cid:90) x ( t )= xx (0)= x D [ z ] D [ p ] D [ x ]e − (cid:98) S/(cid:15) , (6.11)with (cid:98) S ≈ (cid:90) t (cid:20) p ˙ x + qz − λ − h ( x, p, z ) (cid:21) dτ. (6.12) pin coherent states and stochastic hybrid path integrals z by functionally minimizing the action (cid:98) S with respect to z ,noting that q and λ are functions of z via their dependence on φ :0 = δ (cid:98) Sδz ( t ) = dqdφ dφdz z + q − dλdφ dφdz − p [ F ( x ) − F ( x )] − p [ D ( x ) − D ( x )] . (6.13)It then follows from equation (6.8) that q = p [ F ( x ) − F ( x )] + p [ D ( x ) − D ( x )] . (6.14)Hence, the minimized action becomes (cid:98) S = (cid:90) t (cid:20) p dxdt − λ − (cid:0) pF ( x ) + p D ( x ) (cid:1) (cid:21) dτ. (6.15)Finally, defining Λ = λ + pF + p D and substituting for q and λ in equations (6.7)recovers the eigenvalue equation (5.1 a ), and hence (6.11) is equivalent to the pathintegral (5.3).A similar reduction can be carried out for the three-state model with Hamiltonian(4.17). The action (6.16) becomes S = (cid:90) t (cid:20) p dxdt − i(cid:15)φ dzdt + (cid:0) β + (cid:2) − e iφ (cid:3)(cid:1) z + 2 (cid:0) α − (cid:2) − e iφ (cid:3) + α + (cid:2) − e − iφ (cid:3)(cid:1) z (1 − z )+ (cid:0) β − (cid:2) − e − iφ (cid:3)(cid:1) (1 − z ) − h ( x, p, z ) (cid:21) dτ, (6.16)where h ( x, p, z ) = ( pF ( x ) + p D ( x )) z + 2( pF ( x ) + p D ( x )) z (1 − z )+ ( pF ( x ) + p D ( x ))(1 − z ) . (6.17)Generalizing the analysis of the two-state model, we introduce the functions q = β + (cid:2) − e iφ (cid:3) + λ, (6.18 a ) q = β − (cid:2) − e − iφ (cid:3) + λ, (6.18 b ) λ = − α − (cid:2) − e iφ (cid:3) − α + (cid:2) − e − iφ (cid:3) , (6.18 c )and rewrite the action as S = (cid:90) t (cid:20) p dxdt − i(cid:15)φ dzdt + q ( φ ) z + q ( φ )(1 − z ) − λ ( φ ) − h ( x, p, z ) (cid:21) dτ. (6.19)Note that after some algebra one finds that λ, q , q are related according to( q − β + ) r + α + r = λr , (6.20 a ) β + r − ( α + + α − ) r + β − r = λr , (6.20 b ) α − r + ( q − β − ) r = λr . (6.20 c )This can be rewritten in the matrix form (cid:88) m =1 { q m δ n,m + Q nm } r m = λr n , such that q = 0 . (6.21)In the limit (cid:15) → (cid:15) dzdt = dq dφ z + dq dφ (1 − z ) − dλdφ . (6.22) pin coherent states and stochastic hybrid path integrals dz/dt = 0 in the action (6.2) and eliminate the independentvariable φ by imposing the condition dq dφ z + dq dφ (1 − z ) − dλdφ = 0 . (6.23)In particular, e iφ = β − (1 − z ) + 2 α + z (1 − z ) β + z + 2 α − z (1 − z ) . (6.24)We thus obtain the reduced path integral (6.11) with effective action (cid:98) S = (cid:90) t (cid:20) p dxdt + q z + q (1 − z ) − λ − h ( x, p, z ) (cid:21) dτ. (6.25)We can now eliminate z by functionally minimizing the action (cid:98) S with respect to z ,noting that q , q and λ are functions of z via their dependence on φ ( z ):0 = δ (cid:98) Sδz ( t ) = dq dφ dφdz z + 2 zq + dq dφ dφdz (1 − z ) − − z ) q − dλdφ dφdz (6.26) − p [ zF ( x ) + (1 − z ) F ( x ) − (1 − z ) F ( x )] − p [ zD ( x ) + (1 − z ) D ( x ) − (1 − z ) D ( x )] . It then follows from equation (6.23) that q = p ( F − F ) + p ( D − D ) , q = p ( F − F ) + p ( D − D ) . (6.27)Hence, q z + q (1 − z ) − h ( z, x, p ) = − [ pF + p D ][ z + 2 z (1 − z ) + (1 − z ) ]= − [ pF + p D ] , (6.28)and the minimized action is given by equation (6.15). Finally, defining Λ = λ + pF + p D and substituting for q , q and λ into equations (6.20 a )–(6.20 c ) recoversthe eigenvalue equation (5.1 a ) for the three-state model with matrix generator (3.31).
7. Discussion
In this paper we used coherent spin states to derive a new path integral representationof the probability density functional for a stochastic hybrid system evolving accordingto a piecewise SDE. A Langevin equation was obtained in the semi-classical limit,which extended previous diffusion approximations based on a quasi-steady-statereduction. It was also shown how the path integral reduces to a previous version inthe weak noise limit, whose action functional is related to a large deviation principle.In particular, least action paths can be used to determine the most likely paths ofescape from a metastable state.A natural extension of the current work is to explore what happens when thenumber of discrete states is large and the transition rates are n -dependent. Forexample, membrane voltage fluctuations in a neuron may be driven by hundreds ofstochastic ion channels, with the number N ( t ) of open ion channels at time t evolvingaccording to a birth-death master equation [31, 39]: dP n ( n ) dt = (cid:88) m ≥ Q nm P m ( t ) ≡ ω + ( n − P n − ( t ) + ω − ( n + 1) P n +1 ( t ) − [ ω + ( n ) + ω − ( n )] P n ( t ) , (7.1) pin coherent states and stochastic hybrid path integrals ω + ( n, x ) = ( N T − n ) α ( x ) , ω − ( n, x ) = β ( x ) . (7.2)Here N T is the total number of ion channels. Let x ( t ) denote the membrane voltageat time t , which evolves according to the PDMP dxdt = (cid:20) nN T f ( x ) − g ( x ) (cid:21) , (7.3)One could construct a coherent spin- S decomposition of the discrete master equationalong the lines of the two-state and three-state models with 2 S + 1 = N T . This couldthen be used to analyze the resulting stochastic dynamics in the semi-classical limit.However, the expressions become rather cumbersome for large S . On the other hand,it is relatively straightforward to calculate the Hamiltonian Λ in the path integralrepresentation (5.3). For the given ion channel model, the eigenvalue equation (5.1 a )becomes p (cid:18) nN T f − g (cid:19) R n ( x, p ) + ω + ( n − R n − ( x, p ) + ω − ( n + 1) R n +1 ( x, p ) − [ ω + ( n ) + ω − ( n )] R n ( x, p ) , = Λ( x, p ) R n ( x, p ) (7.4)It turns out that the principal eigenvalue can be determined by considering the positivetrial solution [5] R n ( x, p ) = Γ n ( x, p )( N T − n )! n ! . (7.5)Substituting into equation (7.4) yields the following equation relating Γ and Λ : p (cid:18) nN T f − g (cid:19) + nα Γ + Γ β ( N T − n ) − nβ − ( N T − n ) α = Λ . Collecting terms independent of n and terms linear in n yields a pair of equations forΓ and Λ. After eliminating Γ, we obtain a quadratic equation for Λ of the formΛ + σ ( x, p )Λ − h ( x, p ) = 0 , (7.6)with σ ( x, p ) = p [2 g ( x ) − f ( x )] + N T ( α ( x ) + β ( x )) ,h ( x, p ) = p (cid:20) g ( x ) (cid:18) p ( f ( x ) − g ( x )) − N T ( α ( x ) + β ( x ) (cid:19) + N T α ( x ) f ( x ) (cid:21) . One of the roots corresponds to the principal eigenvalue. Elsewhere the reultingHamiltonian system has been used to determine least action paths associated withthe noise-induced from a neuron’s resting state [31, 39].Another possible extension of the current study would be to consider diffusionin a randomly switching environment; one mechanism for switching would bestochastically-gated reactions such as adsorption. Mathematically speaking, thisprocess is an infinite-dimensional version of a stochastic hybrid system, in whichthe piecewise deterministic dynamics is given by a reaction-diffusion equation. Onemethod for analyzing such a system is to discretize space and construct the Chapman-Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybridsystem [9]. One could then use coherent spin-states to construct a path-integralrepresentation of the lattice system. Retaking the continuum limit would then generatea path integral functional for the hybrid reaction-diffusion model. pin coherent states and stochastic hybrid path integrals [1] Berg H C, Purcell E M 1977 Physics of chemoreception. Biophys. J. Phys. Rev. E
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