Noise induced effects

Signals, which are detected by receptors of the biological cell always carry a certain amount of (extrinsic) noise. A signal may consist of chemical as well as physical stimuli (like light, heat, or electro-chemical signals further downstream the signalling cascade), which influence the propensities of reactions in the signalling cascade. Here, we want to demonstrate, that the functional integral description of chemical reactions is very useful in detecting noise induced effects, which sometimes appear to be counter-intuitive, because they improve the cellular response instead of blurring or disrupting it. Such noise-induced effects are well-known in systems described by Langevin and Fokker-Planck equations (see for instance [Garcia-Ojalvo et al., 1992] or [Santos and Sancho, 1999]) ⁸, but are only rarely studied in the context of more complicated chemical reactions. To be specific, let us consider the following simple reaction system

k₁ :A→2A k₂ :A+A→ ∅

k₃ :A→ ∅ The Hamiltonian of this system is given by

H₀ =−k₁(ϕ²−ϕ)η−k₂(1−ϕ²)η−k₃(1−ϕ)η (2.15) The stationary states of this equation are given by

η₁ = 0

8In a biological context the constructive nature of noise is best known from the study of stochastic resonance, which accounts for the phenomenon that correctly tuned fluctuations lead to the enhancement of an external signal [Gammaitoni et al., 1998].

and

η₂ = k₁−k₃

k₂ fork₁ > k₃

The latter state may considered as the active response of the system to signals. Every one of the three propensities could in principle be modified by some signal, depending on the receptor mechanism. Let us first consider the case, where k3 = k +ξ(t) and ξ(t) is Gaussian noise with a very short correlation time τ_ξ as a typical simple model of environmental noise⁹. In the limit τ_ξ →0we get the idealised model ofGaussian white noise, completely characterised by the average

< ξ(t)ξ(t⁰)>_ξ=C₀δ(t−t⁰) (2.17) Averages of observables over internal and external noise may be obtained from the averaged generating function< G(ϕ)>_ξ. Let us therefore consider theξ-average within the functional integral representation. There is a subtle point here, which becomes more obvious when considering the averaging of the time sliced product. In our case it takes on the form

N

Y

k=1

exp[−Hˆ₀δt−∆w_k(1−ϕ)ˆˆ η]

with

∆wk = Z tk

tk−1

ξ(t)dt

Hˆ₀ denotes the noise free part of the Hamiltonian. The∆w_kare independently distributed Gaussian stochastic variables with zero average <∆w_k >= 0 and variance

<(∆wk)² >ξ=C0δt, (2.18) in accordance with Eq.(2.17), so that each factor of the time sliced product can be averaged separately. After inserting the completeness relations, we evaluate the factors

<hϕ_k|exp[−Hˆ₀δt−∆w_k(1−ϕ)ˆˆ η]|ηk−1i>_ξ (2.19) by expanding the exponential neglecting terms of O(δt²). Note, however, that this expansion now requires to keep terms of O(∆w_k²), which after averaging become terms of O(δt). Thus we get

exp[−Hˆ0δt+∆wk(1−ϕ)ˆˆ η] =−

1 + ˆH0δt− C₀

2 (1−ϕ)ˆˆ η(1−ϕ)ˆˆ ηδt

+O(δt²) (2.20)

The result of the averaging may be considered as an addition to the Hamiltonian Hˆ₀. Note, however, that the additional term is not ordered in such a way that all the ϕˆ

9Note that an increase in signal corresponds to a decrease in the degradation ratek₃.

2.2. OPERATOR DESCRIPTIONS OF THE CHEMICAL MASTER EQUATION: SINGLE

COMPARTMENT 27

operators are on the left to all theηˆoperators. Thus we have to reorder this term before we can actually evaluate the expression Eq.(2.19). The correctly ordered Hamiltonian looks as follows:

Hˆ = ˆH₀+C₀

2 (1−ϕ)ˆˆ η−C₀

2 (1−ϕ)ˆ ²ηˆ².

The second term on the r.h.s. of this equation corresponds to a renormalisation (shift) of the net rate ofA-production and it will show up in the rate equation by the replacement

(−k₃+k₁)→

The3^rdterm on the r.h.s. will not show up in the rate equation and may be considered as a modification of the internal chemical noise. It is somewhat counter-intuitive that the effect of the noise is toenhancethe signal (i.e. lead to a decrease of the degradation rate).

Such noise induced effects have been widely studied using Langevin equations. When using this formalism, it is well-known that the renormalisation of rates appears and leads to qualitative changes in the system dynamics for spatially extended systems. ¹⁰ Exam-ples include noise-induced phase transitions, noise-induced front propagation and noise induced generation of patterns. A survey of such phenomena is given in [Sancho and Garcia-Ojalvo, 2000].

The added value of the present approach is not only that it makes it easy to transcribe many of the results obtained for Langevin equations to chemical reaction-diffusion systems described by Master equations. It also gives direct and easy access to noise induced renormalisation due to more complex couplings between the signal and the receptor. The general recipe to establish additional terms due to noisy external signals should be clear by now. If we can split the Hamiltonian into a noise-free part and a coupling of extrinsic (Gaussian white) noise to the system as follows

Hˆ = ˆH₀+ξ(t) ˆH₁,

and then repeat the above steps we end up with a Hamiltonian H˜ = ˆH0− C₀

2

Hˆ₁² = ˆH0+ ∆ ˆHnoise

after averaging overξ. Finally we have to reorderHˆ₁² appropriately, bringing all η oper-ators to the right.

Let us illustrate the procedure for the other two propensities of our reaction system.

If external noise couples to the autocatalytic reactionk₁ →k₁ +ξ(t) we get

∆ ˆHnoise=−C₀

2 ϕ( ˆˆ ϕ−1)ˆηϕ( ˆˆ ϕ−1)ˆη,

10For single compartment systems, the presence of noise prevents an actual phase transition and the modified noise counteracts the renormalisation effect for long times.

which after reordering takes on the form

∆ ˆHnoise=−C₀

2 [ ˆϕ( ˆϕ−1)(2 ˆϕ−1)ˆη+ ˆϕ²(1−ϕ)ˆ ²ηˆ²]

Replacing the operators by numbers and differentiating with respect to ϕ at ϕ = 1 we get the additional terms in the rate equation. Here

∆ ˙η|_noise= C₀ 2 η,

i.e. the same renormalisation as induced by k3 noise. (Note, however, that the addi-tionally induced terms look completely different in these two cases, indicating a different renormalisation of intrinsic noise)

Finally, if the extrinsic noise couples by k2 →k2+ξ(t)we have

∆ ˆH_noise =−C₀

2 (1−ϕˆ²)ˆη(1−ϕˆ²)ˆη.

Reordering this term we get

∆ ˆH_noise =−C₀

2 [(1−ϕˆ²)²ηˆ⁴−4(1−ϕˆ²) ˆϕηˆ³−2(1−ϕˆ²)ˆη²] and this leads to

∆ ˙η|noise = C₀

2 [8η³+ 4η²].

Note that these terms change the global dynamical behaviour of the system qualitatively.

The η³-term leads to an inevitable instability and a run-away behaviour, whereas the η² term shifts the binary decay towards global instability. Thus the system is structurally unstable against noise in the binary decay term. This is also a somewhat surprising result, which shows that certain receptor mechanisms require further elements of the biochemical network to stabilise it against extrinsic noise.