In this article we develop a method to derive the amplitude distribution of stochastic oscillations from the knowledge of the stationary density distribution of a stochastic differential equation (SDE). Here we first give a short outline of our results for which we present the corresponding theorems and proofs in the next sections.

Biochemical networks with stochastic dynamics can be modelled by the Langevin equation 11. For the derivation of the theoretical results, we consider a two-dimensional Langevin equation having an equilibrium point at zero:

with the state vector x ∈ ℛ², the system's dynamics f = (f₁, f₂)^T, where f₁ and f₂ are smooth functions (∈ C^∞), Σ a 2-by-2 matrix of smooth functions and Γ = (Γ₁, Γ₂)^T, where Γ₁ and Γ₂ are uncorrelated, statistical independent Gaussian white noise with zero mean and variance of one.

Let us denote the stationary density distribution of the system (1) by P (x), and the amplitude distribution for the oscillations by P_A(χ), where χ denotes the oscillation amplitude. Under certain assumptions (see following sections) the amplitude distribution P_A(χ) then satisfies

with ν(x) = ||f(x)|| the average speed at state x and (χ) given by

Hereby A is defined as A = {x ∈ ℛ²|⟨ x|⟩ = χ } and is the unit vector in the direction where the amplitude is measured.

The product on the right hand side of (2) represents the steady state flux of reactions through a certain state . Intuitively speaking, is chosen so that most of the realizations going through will also reach their maximal value in a small neighborhood of , so that under certain assumptions (see following sections) it is justifiable to approximate the amplitude of such a realization by the value reached at .

The apparent complicated definition of is a result of the freedom of choice in which direction the amplitude is measured. In a biological system it is obviously important to distinguish whether one measures the oscillations of the concentration of species A or of species B. Sometimes it is experimentally not possible to distinguish between two species, e.g. if B represents the phosphorylated version of a protein A, so that one can only measure the amplitude of [A] + [B]. Additionally, for nonlinear networks, the amplitude distribution may be different depending on whether one measures the amplitude of the oscillations in the positive direction from the steady state or in the negative direction. If one e.g. measures the amplitude distribution of the concentration of the first species x₁ in the positive direction, (3), i.e. = (1, 0)^T, can be simplified to .

To normalize the probability distribution P_A of the amplitude, we have to divide equation (2) through the integral of the probability of all amplitudes:

Simple models may have additional properties allowing us to simplify (2). For linear systems the speed ν((χ)) is directly proportional to ||(χ)|| simplifying (2) to

with given as above.

We want to emphasize here that (2) allows to analytically or numerically calculate the amplitude distribution only from the knowledge of certain properties of the SDE describing the biological system without the necessity to run numerical simulations. The formula yields a good approximation for a wide class of nonlinear problems and can be used to calculate the amplitude distribution even for certain systems having more than two dimensions as we demonstrate in the second example of this paper.

To derive the results presented in the proceeding section we analyze nonlinear SDE systems of order two as given by (1), with an asymptotically stable equilibrium point and damped oscillations in the deterministic part. We restrict our analyis to two dimensional systems, because many higher dimensional systems can be reduced to two dimensions for the purpose of analysing stochastic oscillations. In the derivation we assume w.l.o.g. that we want to compute the amplitude of the oscillations in the positive direction of x₁, i.e. = (1, 0)^T, and that the equilibrium point of the deterministic formulation ( = f(x)) of (1) is at the origin. Both assumptions can easily be satisfied for an arbitrary system by an appropriate coordinate transformation.

We establish an angular phase relationship of the state vector x(t) with respect to a fixed reference vector in the state space 12. An oscillation period of the system (1) is defined as the time during which the angle between the state vector and the fixed reference vector changes by 2π. Further, the amplitude of an oscillation is defined as the maximal value of x₁ during the corresponding oscillation period.

We make the following assumptions on the system (1):

1. There exists a stationary density distribution P: ℛ² → ℛ⁺: x ↦ P (x) for (1) which is sufficiently smooth in x. We demand the curvature of the level curves of this distribution to exist and not to change its sign. For the computation of the amplitude distribution P_A(χ), P (x) may be computed analytically, numerically, or may be obtained by the long term limit of a measurement.

2. The deterministic formulation ( = f(x)) of (1), which is at the state := x(t₀) at time t = t₀ with the probability of the state being P () =: , will approximatively evolve tangential to the level curve of the probability distribution, thus for small Δt the probability of the state at time t + Δt won't have changed significantly (P (x(t₀ + Δt)) ≈ ) This assumption can be easily checked by calculating the Lie derivative of P with respect to f:

which has to be small for almost all x ∈ ² having a non-negligible probability P (x).

3. In addition we require that the average speed ν = ||f(x)|| of (1) does not vanish for almost all x ∈ ℛ² having a non-negligible probability P (x).

Assumption 3 is obviously not satisfied in a small area around the EP where for most systems ν becomes very small. However when the system is very close to the EP, the motion of most realizations can be approximated by a random walk. Thus during the time when the system is in the small neighborhood of the EP it is not justifiable to speak about oscillations anymore until the system leaves the neighborhood again. Away from the EP, ||f(x)|| is significantly bigger than zero for most biological relevant systems so that Assumption 3 is fulfilled.

With the help of these assumptions we are able to formulate a theorem for the calculation of the amplitude distribution of stochastic oscillations. For the formulation of the theorem let us define

i.e. the unit vector in the direction of x₁ is orthogonal to the tangent on the level curve of the probability distribution in (see Figure 1). Because of Assumption 1 there exists exactly one state ∈ for every amplitude χ satisfying = χ. Furthermore we define the following variables, which can be calculated with the knowledge of the system (1) and the stationary probability distribution P:

The variable ν can be thought of as the deterministic speed, L is the standard deviation of the Lie-derivative of P with respect to the right hand side of (1), a the derivative of P in the direction of x₁, and κ the curvature of P with respect to x₂, each evaluated at the states ∈ . The variables are illustrated in Figure 2.

With the help of these definitions we can formulate the following two lemmas and a theorem, which characterizes the amplitude of the stochastic oscillations.

Lemma 1 Assume the Assumptions 1-3 are satisfied. Let Ψ_f be a realization of (1) being at the state ∈ at time t₀. Then the amplitude of Ψ_f during the current oscillation will lie with a probability of 70.8% in the set [, + δx₁] and with a probability of 95.5% in the set [, + ], with δx₁ defined by .

Lemma 2 Assume the Assumptions 1-3 are satisfied. Then the net flux density (x) of realizations at the state x ∈ ℛ² has an absolute value proportional to the product of the probability P (x) and the average speed v(x) of (1) and is tangential to the level curve of the density distribution P at this point.

Lemma 2 is derived by considering the flux of realizations through an infinitesimaly small region around the state x, as illustrated in Figure 3.

Theorem 1 (Distribution of amplitudes) Under the Assumptions 1-3 and if δx₁ as defined in Lemma 1 is small, the probability P_A(χ) of an amplitude χ of the stochastic oscillations of (1) in the direction x₁ is proportional to the absolute value of the net flux density |||| at the state ∈ satisfying = χ.

Our results for the calculation of the amplitude distribution P_A(χ) presented in the previous section (2) directly follow from the application of Theorem 1. The proofs of the respective lemmas and theorem can be found in the appendix.

• To check if the assumptions necessary for the application of Theorem 1 are fulfilled, one can calculate δx₁ according to the formula in Lemma 1. If δx₁ is significantly smaller than for almost all ∈ having a non-negligible probability P (), Theorem 1 may be applied. For a general stochastic differential equation, it is hard to exactly indicate how much smaller δx₁ must be compared to . This highly depends on the structure of the system and also on the accuracy being required for the solution. Nevertheless, as a rule of thumb, the theorem gives good approximations if < 0.2. It is not necessary that this is true for every ∈ , but the calculated density of the amplitude may differ from the real one for the amplitudes χ ∈ for which the corresponding δx₁ is not small enough. Nevertheless the accuracy of the amplitude distribution for other amplitudes is not affected (see proof).

• It may be possible to relax the assumption that the curvature of the level curves does not change its sign globally. Intuitively it seems to be sufficient for most systems that the curvature does not change its sign only locally around the states ∈ , and that the amplitude in one oscillation period reaches its maximum in the area around . However this has to be shown for the system of interest explicitly.

• If the Lie derivative L_fP of the density distribution P is not small enough, the net flux density cannot be approximated to be tangential to the level curves of P anymore. Depending on the size of L_fP this may lead to a significant bias in the calculation of the amplitude distribution for certain systems. In this case, the derivation of the amplitude distribution would be more involved, and also depend on the term , starting from an appropriate modification of (66) in the appendix.

In the following section we apply the algorithm developed in this paper to a realistic example from biochemical signal transduction. A frequent module in many eucaryotic cells from yeast to mammals is the mitogen activated protein (MAP) kinase signaling cascade. MAPK cascades are typically activated by extracellular stimuli such as growth factors, and regulate the activity of various genes, thereby provoking a cellular response to the applied stimulus. Many important cellular functions such as differentiation, proliferation and death are controlled by MAPK cascades 14. MAPK cascades consist of three layers of kinases, where each kinase phosphorylates and thereby activates the kinase on the next layer, as shown in Figure 5. The kinases are named MAPKKK, MAPKK, and MAPK, in the order of activation, and the phosphorylated, active forms are denoted with a star. The basic structure may be complemented by additional feedback interconnections, giving rise to deterministic limit cycle oscillations 15. More recently, oscillations in the MAPK cascade have been determined experimentally in yeast cells, where the oscillatory activity of the MAPK controls periodic changes in cell shape during the mating process 16.

In this section we give a short proof for Lemma 1. We therefore consider the level curve defined by of the density distribution going through the state ∈ .

In the following we argue that there exists a δx₁ such that for a high probability an arbitrarily chosen realization of (1) being at state at time t won't reach a state x with x₁ > + δx₁ until the next oscillation and therefore the measured amplitude will lie in [, + δx₁] with δx₁ positive and small but yet not further determined (see Figure 2). We first calculate the Taylor series expansion for P up to the order of two:

After substituting the definitions made in the equations (8) and utilizing the knowledge that vanishes because of the definition of , setting

, we get

If we stay on a level curve, it must hold that δP = 0 and δx₁ has to be a function of δx₂ locally around . This function is approximated with another Taylor series expansion up to order two as

with k₁, k₂ yet unknown constants. By substituting (43) in (42) we get

Because the coefficients of δx₂ and must vanish independently for (44) to hold, it follows that k₁ = 0 and k₂ = , giving

This means that we may approximate a level curve of the density distribution locally by a parabola. For small deviations and small times t it is possible to make some approximations for (1). First we may approximate Σ (x) in an area around by the constant

We can rewrite the norm of the first row elements of by

which can be easily validated by the definitions of L and a given in the equations (8). We furthermore approximate the second element of f(x) in an area around by

Putting everything together we get a one dimensional Wiener process for the movement of (1) in the direction of x₂ for small times Δt:

with Δx(t) = x(t) - . We approximate the average displacement in the direction of x₁ by combination of the average displacement of this Wiener process (⟨ x₂(t)⟩ = νt) with equation (45):

Because the stochastic part of (49) may be neglected (see Assumption 4), we can get a linear one dimensional stochastic equation for the movement of (1) in the direction of x₁ for small times t:

It can be shown [10, p. 129 ff.] that the variance of (51) evolves as

We may think of the solution of (51) as a growing Gaussian distribution moving along the trajectory of the deterministic system (see Figure 2). We now determine the maximal value of Δx₁(t) that a realization of (1) starting at may have if it evolves in an area around the average displacement determined by the standard deviation . The maximal value of Δx₁(t) is time dependent and grows in the beginning due to the stochastic part of (51), but afterwards shrinks due to the deterministic part. It reaches its maximal value when the time derivatives of the mean value and of the standard deviation have the same absolute value:

We can now calculate δx₁ to

For a Gaussian distribution it is true that 68.3% of all realizations stay in an area around the average displacement determined by the standard deviation. This means that only of the realizations reach an amplitude greater than + δx₁ after passing . Because it is also possible for a realization to have its maximal value in the direction of x₁ before passing , the overall probability for a realization to have an amplitude lower than + δx₁ is

In the same way it can be shown that 95.5% of all realizations have an amplitude lower than + .

Assume a small unit cell with the edge lengths dr and dφ with its center being x and the level curve of P through x given and going through the center of the edges with the lengths dr (see Figure 3). If dr and dφ are small, we may assume the density distribution P and the values f(x) and Σ(x) of (1) as constant inside the unit cell. Due to the small size of the Lie-derivative of P along f, the movement normal to the level curve is dominated by the stochastic part of (1). The net fluxes ϕ₁ and ϕ₃ trough the edges 1 and 3 are vanishing and therefore the overall flux is tangential to the level curve. The net flux ϕ₂ (ϕ₄), trough the edges 2 (the edge 4) can be determined by integration (see, for example, [25, p. 622]):

By letting dr and dφ go to zero the absolute value of the net flux density ||(x)|| follows:

Because the density distribution P is smooth and the curvature of its level curves does not change its sign, there exists exactly one state ∈ with = χ for every χ ∈ (0, ∞). is a smooth simple curve in the state space which can be parameterized by

with an unknown but smooth function in χ. We can get a term proportional to the average number of oscillations of all realizations per time unit if we determine the total net flux of realizations ϕ_t through this curve:

with the normalized normal vector on in . We may determine the probability P_A(χ) of the amplitude χ ∈ (0, ∞) by multiplying the flux density of realizations with Ω (, χ), the probability that a oscillation of a realization contributing to the flux density at has the amplitude, and integrating over all ∈ :

From the condition we get

Because of Lemma 2 we know that the flux density is directed along the x₂-axis and has the absolute value P () ν(). Therefore we can calculate (64) to

Although we do not know the exact shape of Ω (, χ), which may be complicated to calculate, we know a lower and an upper bound for Ω. The lower bound is given due to the fact that Ω must be zero for χ <, because if (1) goes through the state , it has at least the amplitude of this state. An upper bound for almost all realizations is given by Lemma 1 with + . We may therefore conclude that Ω is approximately zero for χ being outside of the interval [, + ], which simplifies (67) to

If δx₁ is small, we can assume P and ν being approximately constant in the integration interval, which simplifies (68) to

For small δx₁ the integral of Ω over is approximately 1. The result of this last approximation is given by Theorem 1.