Institute for Systems Theory and Automatic Control, Universität Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany

Abstract

Intrinsic noise is a common phenomenon in biochemical reaction networks and may affect the occurence and amplitude of sustained oscillations in the states of the network. To evaluate properties of such oscillations in the time domain, it is usually required to conduct long-term stochastic simulations, using for example the Gillespie algorithm. In this paper, we present a new method to compute the amplitude distribution of the oscillations without the need for long-term stochastic simulations. By the derivation of the method, we also gain insight into the structural features underlying the stochastic oscillations. The method is applicable to a wide class of non-linear stochastic differential equations that exhibit stochastic oscillations. The application is exemplified for the MAPK cascade, a fundamental element of several biochemical signalling pathways. This example shows that the proposed method can accurately predict the amplitude distribution for the stochastic oscillations even when using further computational approximations.

**PACS Codes: **87.10.Mn, 87.18.Tt, 87.18.Vf

**MSC Codes: **92B05, 60G10, 65C30

1. Introduction

Oscillations are a widely occurring phenomenon in the dynamics of biological systems. On the intracellular level, oscillations occur for example in the activity of various genes or signalling proteins. To gain more insight into the processes related to these oscillations, mathematical models for the underlying biochemical and genetic networks are commonly constructed. Such models have proven helpful in connecting the underlying biochemistry with the temporal characteristics of emerging oscillations and the associated biological function. Examples for this type of results include the intracellular circadian clock

In dynamical models for biological systems, oscillations are due to limit cycles or more complex attractors in deterministic frameworks, or may emerge from various stochastic effects in stochastic frameworks. Concerning the biological function, one would not expect that it makes a significant difference whether oscillations are due to a deterministic limit cycle or arise from stochastic effects. For the biological function, rather the temporal characteristics of oscillations are relevant, such as their frequency or amplitude. For deterministic limit cycle oscillations, these characteristics are easily computed by a numerical simulation of the model. For stochastic models, computing the temporal characteristics of oscillations is much more involved. Two approaches are common for solving stochastic systems. In the first approach, the so called chemical master equation (CME) is used

In this paper, we develop a new method to compute the amplitude distribution for systems exhibiting stochastic oscillations. We thereby focus on systems where the deterministic part has a weakly stable equilibrium point (EP), typically with damped oscillations, and the stochastic effects induce sustained oscillations around this EP. In such systems, we can distinguish two mechanisms by which stochastic oscillations occur. For the first mechanism, the system needs to have the property that a certain small perturbation away from the equilibrium point leads to a large excursion in the state space before the system returns to the proximity of the EP

For the second mechanism, it is of interest to compute the amplitude distribution of the stochastic oscillations. Several methods to solve this problems were developed previously. In

In this article we focus on biochemical systems that can be modeled as a set of interconnected, possibly nonlinear, stochastic differential equations (SDEs). With the help of the Fokker-Planck equation (see e.g. [3, p. 193 ff.] or [10, p. 120 ff.]), we calculate the stationary density distribution in the state space. We provide a theorem allowing to calculate the amplitude distribution with only the knowledge of the density distribution and the corresponding SDE. The theorem allows not only to consider linear, but also a wide class of non-linear systems and therefore makes it possible to analyze not only systems with oscillations being of low amplitude compared to average concentrations but also of intermediate and high amplitudes. Although stochastic oscillations are a mainly two dimensional effect, we show that they also occur in higher order systems and give an example on how to analyze them then. As far as we are aware of, this paper introduces for the first time a method to analytically analyse higher order possibly non-linear systems being affected with intermediate to high amounts of noise, and thereby showing stochastic oscillations.

The paper is structured as follows. To give an easily accessible introduction to the material, we first present our results on the calculation of the amplitude distribution of stochastic oscillations and afterwards state the underlying assumptions and theorems. Then we explain the application of our theory by calculating the amplitude distribution of a simple example, the damped harmonic oscillator in the presence of additive noise, to get a first insight into the reasons for stochastic oscillations. We want to remark that this example simplifies, due to its structure, the necessary calculations significantly. As a more realistic example, we discuss oscillations in the MAP kinase cascade with an incorporated negative feedback with limited amounts of entities of each molecular species. For this system, the stationary probability distribution can be estimated by a linear approximation, or it can be computed numerically. We compare the amplitude distributions predicted by our method, based on these two approaches, to an "experimental" amplitude distribution obtained from a long-term stochastic simulation.

2. Results and Discussion

2.1. Amplitude Distribution of the Stochastic Oscillations

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

with the state vector **x **∈ ℛ^{2}, the system's dynamics **f **= (_{1}, _{2})^{T}, where _{1 }and _{2 }are smooth functions (∈ ^{∞}), **Σ **a 2-by-2 matrix of smooth functions and **Γ **= (Γ_{1}, Γ_{2})^{T}, where Γ_{1 }and Γ_{2 }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 **x**), and the amplitude distribution for the oscillations by _{A}(χ), where χ denotes the oscillation amplitude. Under certain assumptions (see following sections) the amplitude distribution _{A}(χ) then satisfies

with **x**) = **f**(**x**)**x **and

Hereby **x **∈ ℛ^{2}|⟨ **x**|

The product on the right hand side of (2) represents the steady state flux of reactions through a certain state

The apparent complicated definition of _{1 }in the positive direction, (3), i.e. ^{T}, can be simplified to

To normalize the probability distribution _{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

with

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.

2.2. Derivation of the Results

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 _{1}, i.e. ^{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**(_{1 }during the corresponding oscillation period.

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

1. There exists a stationary density distribution ^{2 }→ ℛ^{+}: **x **↦ **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 _{A}(**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**(_{0}) at time _{0 }with the probability of the state **x**(_{0 }+ Δ**f**:

which has to be small for almost all **x **∈ ^{2} having a non-negligible probability **x**).

3. In addition we require that the average speed **f**(**x**)**x **∈ ℛ^{2 }having a non-negligible probability **x**).

Assumption 3 is obviously not satisfied in a small area around the EP where for most systems **f**(**x**)

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 _{1 }is orthogonal to the tangent on the level curve of the probability distribution in

Definition of

**Definition of **. The solid black lines are the level curves of

The variable _{1}, and _{2}, each evaluated at the states

Visualization of the variables

**Visualization of the variables**. Visualization of the variables needed to calculate the amplitude distribution of stochastic oscillations.

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 **_{f }be a realization of (1) being at the state _{0}. Then the amplitude of Ψ_{f }during the current oscillation will lie with a probability of 70.8% in the set _{1}] _{1 }defined by

**Lemma 2 ****x**) **x **∈ ℛ^{2 }**x**) **x**)

Lemma 2 is derived by considering the flux of realizations through an infinitesimaly small region around the state

Flux through unit cell

**Flux through unit cell**. The fluxes through the edges of an infinitesimal small unit cell around a state **x **∈ ^{2}.

**Theorem 1 (Distribution of amplitudes) **_{1} as defined in Lemma 1 is small, the probability P_{A}(_{1 }

Our results for the calculation of the amplitude distribution _{A}(

2.3. Remarks

• To check if the assumptions necessary for the application of Theorem 1 are fulfilled, one can calculate _{1 }according to the formula in Lemma 1. If _{1 }is significantly smaller than _{1 }must be compared to _{1 }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

• If the Lie derivative _{f}_{f}

2.4. The Damped Harmonic Oscillator in the Presence of Additive Gaussian White Noise

2.4.1. Equations and Properties of the System

In this section we give a motivational example for the application of Theorem 1 by discussing the behavior of the damped harmonic oscillator in the presence of additive Gaussian white noise. The oscillator satisfies the following Langevin equation:

with Γ_{i}, ^{+ }and

It is easy to show that the deterministic system (

are at λ_{1,2 }= -

2.4.2. Calculation of the Density Distribution

For _{i},

with **x**) and _{i, j }(**x**) **B **may be calculated to

with **Σ **the matrix of the noise terms of (1). We call (11) linear if ∀_{i}**x**) is a linear function in **x **and ∀_{i, j}(**x**) is constant. Then (11) can be simplified to

For the system (9), _{i, j }are given by the system matrix _{i, j }are the elements of the matrix **B **= ^{2}_{2}, where _{2 }is the second order identity matrix. To get the stationary density distribution, we set

with ⟨ **x **⟩ the mean value of **x **and **Ξ **the matrix of the second moments of P, which is the solution of the equation **AΞ **+ Ξ**A**^{T }+ **B **= 0. In system (9), ⟨ **x **⟩ = **0 **and Ξ may be calculated to

2.4.3. Determination of the Amplitude Distribution

To get the amplitude distribution _{A}(^{T }for all

After some calculations we obtain

For _{1 }is getting small compared to

Due to the linearity of the oscillator (9) and the symmetry of the harmonic oscillator, we can determine the amplitude distribution by utilizing formula (5) as

For _{A }is plotted over the amplitude

Amplitude distribution of the oscillations of the damped harmonic oscillator

**Amplitude distribution of the oscillations of the damped harmonic oscillator**. Amplitude distribution of the oscillations of _{2 }of the damped harmonic oscillator (9) in the presence of white noise with autocorrelation

This easy example was discussed to give an insight into the reasons for stochastic oscillations. In the next section, we show the practical applicability of our algorithm by predicting the amplitude distribution of a complex biochemical system and therewith give an example of a more biologically relevant application for the results of the paper.

2.5. Oscillations in the MAP Kinase Signaling Cascade

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

Schematic view on the MAP kinase signaling cascade

**Schematic view on the MAP kinase signaling cascade**. Schematic view on the MAP kinase signaling cascade. The species with the stars (*) are the phosphorylated versions of the species without stars. MAPKKK* catalyzes the phosphorylation of MAPKK, MAPKK* in turn catalyzes the phosphorylation of MAPK. A negative feedback is given by the repression of the activation of MAPKKK by MAPK*.

2.5.1. The Deterministic Model

For this example, we use a basic ODE model of the MAP kinase signaling cascade. The model contains a negative feedback interconnection from the last kinase MAPK to the activation of the first kinase MAPKKK

Denote _{1}, _{2 }and _{3 }the concentrations of MAPKKK*, MAPKK* and MAPK*, respectively. Using three conservation relations

we get the deterministic description of the system using mass balancing with Michaelis-Menten reaction kinetics:

with the reaction rates given by

The parameter sets for the phosphorylation (_{ij}) and for the dephosphorylation (_{kl}) are given in Table

Parameter Set for the MAP Kinase Signaling Cascade

**Parameter**

**Value**

**Unit**

_{11}

0.256

1/s

_{12}

0.0872

1/nM

_{13}

0.1144

1/nM

_{21}

0.4229·10^{-2}

1/s

_{22}

0.0347

1/nM

_{31}

0.2049·10^{-2}

1/s

_{32}

0.0243

1/nM

_{11}

0.1003

1/s

_{12}

0.1428

1/nM

_{21}

0.1218

1/s

_{22}

0.0794

1/nM

_{31}

0.1343

1/s

_{32}

0.0167

1/nM

The biochemically relevant equilibrium point **x**_{0 }of the system is computed as

We can determine the local stability of this EP by analyzing the Jacobian

and thus the deterministic system is locally asymptotically stable.

2.5.2. The Stochastic Model

For validation purposes and to get a first insight into the system's dynamics, we did stochastic simulations of the biochemical network model of the MAPK cascade as described by (19). We used the stochastic simulation software Dizzy by the Institute for Systems Biology

where _{A }is the Avogadro constant. We provide a typical plot of the oscillations of _{1}, _{2 }and _{3 }against the time in Figure

Oscillations of the MAP kinase signaling cascade

**Oscillations of the MAP kinase signaling cascade**. Plot of the oscillations of the MAP kinase signaling cascade. The plot is the result of a stochastic simulation via the Direct Gillespie approach with Dizzy.

As discussed in

with Γ_{i, f }and Γ_{j, b}, _{A }the Avogadro constant and **x **and don't change significantly in an area around the EP, so that we can approximate them with their steady state values. Afterwards we can combine each function's noise terms by adding their variances

with

2.5.3. Transformation and Model Reduction

The theoretical results in this paper have been developped for two-dimensional systems only. In higher order systems stochastic oscillations may appear on a two dimensional manifold in the state space. Such a system can be reduced to an order two system by computing the slow manifold, making use of a time scale separation

with _{1 }∈ **z _{slow }**= (

The transformation needs to be done in such a way that the absolute values of the real parts of the eigenvalues of _{fast }are much larger than these of **A**_{slow}. To this end, we compute the eigenvectors corresponding to the eigenvalues (22a)-(22b) as

The two eigenvectors with non-zero imaginary parts, _{2 }and _{3}, are both eigenvectors corresponding to eigenvalues having real parts with small absolute values. Therefore it is straightforward to define the desired transformation as

with the transformation matrix **T **= (**e**_{1}, **e**_{2 }+ **e**_{3}, (**e**_{2 }- **e**_{3})·

Because all of the eigenvalues (22a)-(22b) have non-zero real parts, the Hartman-Grobman theorem states that there exists a local transformation **z **= **H**(

Because in the **z**-system there exists a slow manifold, too (see _{1 }= _{2}, _{3}) which describes the dependence of the states on the manifold. We approximate this function with a truncated Taylor series expansion:

We substitute (31) in _{z,1 }and can therewith calculate the coefficients _{j, k }(see Table **x**-coordinates is shown in Figure

Parameter Set of the Slow Manifold of the MAP Kinase Signaling Cascade

**Parameter**

**Value**

**Parameter**

**Value**

**Parameter**

**Value**

_{0,0}

0

_{0,2}

0.01800

_{4,0}

7.327·10^{-8}

_{1,0}

0

_{3,0}

-8.476·10^{-6}

_{3,1}

-8.191·10^{-7}

_{0,1}

0

_{2,1}

3.625·10^{-5}

_{2,2}

-8.162·10^{-7}

_{2,0}

0.003604

_{1,2}

2.425·10^{-4}

_{1,3}

3.951·10^{-6}

_{1,1}

0.01102

_{0,3}

5.750·10^{-4}

_{0,4}

1.785·10^{-5}

Slow Manifold of the MAP kinase signaling cascade

**Slow Manifold of the MAP kinase signaling cascade**. Slow manifold of (20) calculated up to the order of four. The black curve is a representative trajectory of the deterministic system.

The slow manifold is attractive enough so that realizations of the stochastic system (25) will stay close to it. Due to this, it is sufficient to just take the oscillations on the slow manifold into account and therewith simplify the problem to two dimensions. We substitute the formula of the slow manifold (31) in the differential equations for _{2 }and _{3 }(29) and get a two dimensional reduced description of the system

to which Theorem 1 can be applied.

2.5.4. Calculation of the Density Distribution

As a first approach to obtain the stationary density distribution **x**), we make use of a linear approximation of the reduced system (32) around the EP. Taking this approach allows us to evaluate how well our method works with such an approximation, where the stationary density distribution can be obtained with minimal computational effort. For other systems, or if a high precision of the result is required, it could however also be necessary to solve the nonlinear Fokker-Planck equations numerically.

The linear approximation of the reduced system (32) is given by

with the system matrix

and **Γ **the vector of the disturbances Γ_{i},

From the coordinate transformation **Σ**_{η }is determined by

where diag(_{i}) is a diagonal matrix with the diagonal elements _{i }being the standard deviations of the stochastic terms in the system (25). For the system (33) with **Σ**_{η }is obtained as

Following the same approach as in the previous example, the stationary density distribution is obtained as

with

for a cell with a volume of

Linear distribution of the state of the MAP kinase signaling cascade

**Linear distribution of the state of the MAP kinase signaling cascade**. Linear density distribution of the MAP kinase signaling cascade on the slow manifold as calculated in (37). Because we linearized the system, the distribution is Gaussian.

2.5.5. Determination of the Amplitude Distribution

With the preliminary work of the preceding sections we are now able to calculate the amplitude distribution according to Theorem 1. We consider the states **x**_{χ }= (^{T }with

The probability of the amplitude

under the constraint that **z**_{χ }has to lie on the slow manifold. This corresponds to that **z**_{χ }has to satisfy (31), which gives us a dependency of

The resulting amplitude distributions in each of the three original coordinate directions _{i},

Linear amplitude distribution of the oscillations of the MAP kinase signaling cascade

**Linear amplitude distribution of the oscillations of the MAP kinase signaling cascade**. Linear amplitude distribution of the oscillations of the MAP kinase signaling cascade. The solid curve corresponds to data determined by stochastic simulation of the nonlinear system (24), the dashed curve is the prediction according to the calculations in this section.

As can be seen in Figure _{1}], whereas we estimate its value by the lower end _{1}. Further discussion of this point can be found in the conclusions.

We also want to mention that we compare the results obtained from our method to stochastic Gillespie simulations, and not to realizations of the Langevin equation (24). The reason herefor is that Gillespie simulations better predict the behavior of biochemical networks and are thus the method of choice. However in the derivation of the amplitude distribution we approximated the stochasticity of the system using white noise terms. The results of both methods to describe the intrinsic noise of the system may under certain circumstances lead to different results, which may be a further explanation of the small bias between the measured and the theoretical predicted amplitude distribution in Figure

2.5.6. Numerical Approach

Sometimes it is not adequate anymore to analyze the linearized system and one has to analyze the original nonlinear one. Although there might be several special cases where the probability distribution

To calculate the density distribution of the nonlinear system we have to solve the nonlinear Fokker-Planck equation (11). We therefore applied the algorithm developed in

Nonlinear Distribution of the state of the MAP kinase signaling cascade

**Nonlinear Distribution of the state of the MAP kinase signaling cascade**. Density distribution of the MAP kinase cascade on the slow manifold (31).

From the density distribution **x**), we can compute the amplitude distribution _{A}(_{A}(_{2 }for peak amplitudes below the equilibrium point values, i.e. ^{T}, because then the effect of the nonlinearities is even higher than in the other direction. The reason for this is quite simple: The amount of entities of each species is not allowed to become negative. As can be seen in Figure

Nonlinear amplitude distribution of the oscillations of the MAP kinase signaling cascade

**Nonlinear amplitude distribution of the oscillations of the MAP kinase signaling cascade**. Nonlinear amplitude distribution of the oscillations of the MAP kinase signaling cascade in the negative _{2}-direction. The solid curve corresponds to data experimentally measured, the dashed red curve is the prediction according to the calculations in this section and the dash-dotted green curve is the outcome we get by calculating the amplitude distribution of the linearized system.

3. Conclusion

We introduce a method to determine the amplitude distribution of a wide class of linear and nonlinear stochastic systems given by (1), which display sustained stochastic oscillations. The method is applicable to systems where a stationary density distributions exists and can be computed either analytically or numerically.

The method is based on computing the flux density of realizations for the states where the tangent on the level curve of the density distribution is normal to the direction in which the oscillations are measured. We showed that under certain conditions this flux density is directly proportional to the probability of an oscillation with an adequate amplitude to occur. Our results can be used in the analysis of systems being influenced by strong internal or external noise, as we often find them in biophysical problems.

As already discussed at the end of the second example, for certain systems the calculated amplitude distribution may contain a small bias depending on the exact structure of the system and on how well the the assumption necessary for the application of our method are satisfied. For the wide class of nonlinear systems analyzed we could obtain as a result that realizations being at state _{1}] as stated in Lemma 1. By requiring _{1 }to be small we justified to approximate the amplitude by _{1}] can be calculated or estimated, it would be possible to reduce the bias for systems where _{1 }is small but not negligible. Such an extension would lead to a more refined approximation of the amplitude distribution.

Stochastic oscillations may occur in biological systems not only as a disturbing side effect, but also in a constructive manner

Appendix

Proof for Lemma 1

In this section we give a short proof for Lemma 1. We therefore consider the level curve defined by

In the following we argue that there exists a _{1 }such that for a high probability an arbitrarily chosen realization of (1) being at state **x **with _{1 }> _{1 }until the next oscillation and therefore the measured amplitude will lie in [_{1}] with _{1 }positive and small but yet not further determined (see Figure

After substituting the definitions made in the equations (8) and utilizing the knowledge that

If we stay on a level curve, it must hold that _{1 }has to be a function of _{2 }locally around

with _{1}, _{2 }yet unknown constants. By substituting (43) in (42) we get

Because the coefficients of _{2 }and _{1 }= 0 and _{2 }=

This means that we may approximate a level curve of the density distribution locally by a parabola. For small deviations and small times **Σ **(**x**) in an area around

We can rewrite the norm of the first row elements of

which can be easily validated by the definitions of **f**(**x**) in an area around

Putting everything together we get a one dimensional Wiener process for the movement of (1) in the direction of _{2 }for small times Δ

with **Δx**(**x**(_{1 }by combination of the average displacement of this Wiener process (⟨ _{2}(

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 _{1 }for small times

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 _{1}(_{1}(

We can now calculate _{1 }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 _{1 }after passing _{1 }before passing _{1 }is

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

Proof for Lemma 2

Assume a small unit cell with the edge lengths **x **and the level curve of **x **given and going through the center of the edges with the lengths **f**(**x**) and **Σ**(**x**) of (1) as constant inside the unit cell. Due to the small size of the Lie-derivative of _{1 }and _{3 }trough the edges 1 and 3 are vanishing and therefore the overall flux is tangential to the level curve. The net flux _{2 }(_{4}), trough the edges 2 (the edge 4) can be determined by integration (see, for example, [

By letting **x**)|| follows:

Proof for Theorem 1

Because the density distribution

with _{t }through this curve:

with _{A}(

From the condition

Because of Lemma 2 we know that the flux density _{2}-axis and has the absolute value

Although we do not know the exact shape of Ω (

If _{1 }is small, we can assume

For small _{1 }the integral of Ω over