Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, USA

Department of Electrical and Computer Engineering, Purdue University, West Lafayette, USA

Department of Mathematics, Purdue University, West Lafayette, USA

Abstract

The Steady State (SS) probability distribution is an important quantity needed to characterize the steady state behavior of many stochastic biochemical networks. In this paper, we propose an efficient and accurate approach to calculating an approximate SS probability distribution from solution of the Chemical Master Equation (CME) under the assumption of the existence of a unique deterministic SS of the system. To find the approximate solution to the CME, a truncated state-space representation is used to reduce the state-space of the system and translate it to a finite dimension. The subsequent ill-posed eigenvalue problem of a linear system for the finite state-space can be converted to a well-posed system of linear equations and solved. The proposed strategy yields efficient and accurate estimation of noise in stochastic biochemical systems. To demonstrate the approach, we applied the method to characterize the noise behavior of a set of biochemical networks of ligand-receptor interactions for Bone Morphogenetic Protein (BMP) signaling. We found that recruitment of type II receptors during the receptor oligomerization by itself doesn't not tend to lower noise in receptor signaling, but regulation by a secreted co-factor may provide a substantial improvement in signaling relative to noise. The steady state probability approximation method shortened the time necessary to calculate the probability distributions compared to earlier approaches, such as Gillespie's Stochastic Simulation Algorithm (SSA) while maintaining high accuracy.

Introduction

Many biological networks exhibit stochasticity due to a combinatorial effect of low molecular concentrations and slow system dynamics. One important biological context where stochastic events likely have a large impact is the Bone Morphogenetic Protein (BMP) signaling pathway. BMPs make up the largest subfamily of the Transforming Growth Factor-

Stochastic regulation can negatively impact the robustness of the system

The Chemical Master Equation (CME) describes the dynamics of the probability distribution of a species of chemical reactions. Precisely, the CME captures the rate of change of probability that a system will be in state **X **at time ^{N}

In the analysis of stochastic biochemical networks, steady state probability distributions for each species in the system are determined to measure variability about the deterministic steady state. The deviation around the solution contributes to stochastic noise that can be quantified by measuring the coefficient of variation

Frequently, Monte-carlo based simulation approaches

In order to ameliorate the computational cost and complexity, we present a method here to approximate the steady state probability distribution by 1) reducing the system's state-space to a finite dimension using truncated state-space method

The usefulness of the proposed method is illustrated by considering the example networks of BMP signaling, as described earlier in

Unlike SBPs, which tend to improve the signal to noise ratio, we did not see a significant change in stochastic variability with inclusion of Type II receptors. This result supports a previous assumption made in

Methods

Proposed approximation method

The Chemical Master Equation (CME), which is a set of first order differential (ODE) equations, demonstrates loss and gain of probabilities of discrete states of a system _{1}, _{2}, . . . ,_{N}_{1}, _{2}, _{N}_{μ}_{n} molecules_{n}**X **= [_{1}(_{2}(_{N}_{n}

Two other quantities are further required to construct the system: 1) a state-change vector _{μ }_{μ}_{µ }_{μ }_{μ }_{1μ}, _{2μ}, . . . ,_{Nμ}^{T}_{nμ }_{n}_{μ }**X**,

Here, [(**X **+_{μ}**X **+_{μ }**X**, **X **∈ Ω is given by taking Ω to be the non-truncated space: Ω = ℕ^{N}

In Eq.1, _{μ }**X** to any other state, and _{μ }

where P is the probability distribution **X**, **X **= [_{1}, _{2}, . . . , _{N}_{ss}

We assume that the deterministic steady state (SS) is unique. The non-truncated state-space Ω can be replaced with a truncated state-space **X**,

where _{i }_{i }

The truncated state-space representation implies that given some _{i }_{i }_{ss }**X**) is approximated to within

For an optimal SS probability approximation,

where

Instead of finding the eigenvector, which can be an ill-conditioned problem when there are nonzero eigenvalues close to 0, we translate the problem to a well-conditioned system of linear equations as follows.

We first evaluate the deterministic steady state (_{0}) of the system, and then select state _{0} of the discrete system closest to this deterministic steady state, where _{0 }= _{0}). Taking _{0}) is among the largest entries of _{0} denoted by

Then

where ^{th }

In Eq.7,

Application

In order to demonstrate the usability of the proposed steady state probability approximation method, we present here two example networks (example network 1 and 2) from Bone Morphogenetic Protein(BMP) mediated signaling, and characterize the stochastic behavior of the systems. In the example network 1, we consider the role of a specific extracellular protein, Crossveineless-2(Cv-2), which is part of a class of proteins known as Surface-associated BMP-binding Proteins (SBPs)

In the second example network, we consider a model simplification strategy as used in

Background

During embryonic development, positional information is transduced by morphogens to underlying cells that respond to the concentration gradient of morphogen and eventually differentiate into distinct cell types

BMP regulation occurs at many points along the pathway, and a lot of focus has been on identifying and understanding how the ligand activity is regulated in the extracellular environment by secreted binding proteins. These include molecules such as Cv-2, HSPGs, among other reviewed in

Example networks

In many biochemical networks, where dynamics of the intermediate interactions of different species (proteins) and molecular complexes are largely unknown, screening plays a significant role in the classification of dynamics-dependent network behavior. For example:

1. In a biochemical network where a class of secreted, surface-associated BMP binding proteins (SBPs) such as, Crossveinless-2 (Cv-2, node D as in Figure

Example network of BMP signaling

**Example network of BMP signaling**. BMP signaling is mediated by

- B

- R

2. In the patterning modeling of BMP signaling pathways, it is often argued as a simplification strategy that omitting the step of recruitment of a Type II receptor to a bound BMP:Type I receptor complex doesn't affect the outcomes of patterning models

In these systems, we apply our SS probability approximation method to evaluate the probability distribution of different species and calculate the mean (

BMP-signaling regulation by SBPs

Signaling network

The single-cell local stochastic model that includes extracellular BMP(A), receptors (B), and SBPs such as, Cv-2 (D) with biochemical interactions, rate parameters, and connectivity is based on the network shown in Figure

Out of all complexes (C =

- B

- R

- B

- R

The simplified model as obtained from reactions _{1} to _{10}, has 5 species {_{1}, _{2}, . . . , _{5}} and is described completely by a total of 8 different chemical reactions. Time evolution of all species quantities are specified by a state vector **X(t) **= [_{1}(_{2}(_{5}(^{T }_{μ }_{1} is [-1 +1 0 0 0]^{T }_{0} using Newton's method as incorporated in SBTOOLBOX2

**Algorithm 1 **Evaluate steady state (SS) distribution:

**Require: **Unique deterministic SS solution _{0}

1. Reaction Networks with _{1}, . . . , _{N}

2. Choose: _{0},

3. Solve: ODE for steady state(SS) = _{0} and find discrete state _{0} closest to _{0}, where _{0} = _{0}).

4. Initiate, _{i}, β_{i}

5. Determine: Truncated state-space

6. Determine: Column j of _{0}

7. Form

8. Solve:

9. Find

10. Verify:

**if **_{i }_{i}_{i }_{i }**then**

_{i ← αi }- γ

_{i }_{← }
_{i }+ γ

Return to 5

**end if**

In the algorithm, the values of _{0}, _{0} favor a larger

Simulation and discussion

The binding kinetics between BMPs (species A, Figure _{±s}, ^{-1 }^{1}] ^{-1}^{-1} for the forward rates and [10^{-3 }^{0}] ^{-1} for the reverse reaction rates. This produces a parameter grid that contains a total of 625 different parameter vectors.

For an appropriate comparison of the noise attenuation both in the presence and in the absence of

In order to quantify noise in the system we measured the coefficient of variation _{Cv}_{2 ≠ 0 }> Λ_{Cv}_{2 = 0}) and is valid for the range of Cv-2 values considered in the screen (for a detailed discussion on this, interested readers can refer

Screening result

**Screening result**. **a, b, c**) Coefficient of variation **d) **Histogram of noise change for all 625 parameter sets demonstrates that the majority of solutions result in noise attenuation.

To clarify Cv-2 action further, we calculated percentage change in the amplitude using

During the simulation, the kinetic rate constants for the intermediate complex BMP:receptor:Cv-2 (node Z, Figure

Kinetic rates, Figure 2(a,b,c)

**Figure**

**k _{3 }(molecule^{-1 }sec^{-1})**

**k _{-3} (sec^{-1})**

**k _{4} (molecule^{-1 }sec^{-1})**

**k _{-4} (sec^{-1})**

2a

1.3282

0.0100

1.3282

0.0100

2b

0.0133

1.0000

0.1328

0.0100

2c

0.1328

1.0000

0.4200

1.0000

Analysis of Type II receptor recruitment process

In the signaling network shown in Figure _{1}) receptors can happen in two different ways: 1) BMP binds with Type I (=_{1}) first and subsequently, recruits Type II receptors to form a tripartite complex BMP:Type I: Type II (_{1}_{2}), and 2) BMP directly interacts with Type I and Type II separately, and an intermediate state forms a tripartite BMP:Type I: Type II complex. In both situations, BMP:Type I:Type II complex (B_{1}_{2}) is the sole signaling complex responsible for the activation of downstream pathways.

Network cases for Type II recruitment analysis in context of Drosophila melanogaster development

**Network cases for Type II recruitment analysis in context of Drosophila melanogaster development**. **Case I) **Recruitment of Type II is overlooked here and it imitates the simplified model used in previous studies. In this type of network, BMP:Type I complex (B_{1}) acts as the sole signaling complex. **Case II) **Upon the formation of a BMP: Type I complex, subsequent recruitment of Type II receptor is considered here. But a direct interaction between BMP and Type II receptor doesn't happen in the network. Here, a tripartite complex BMP:Type I:Type II (B_{1}_{2}) activates the downstream pathways. **Case III) **Similar to Case II, but with an exception that a direct interaction between BMP and Type II receptor is allowed to form a BMP:Type II complex (_{2}) by changing _{1} to _{2}. The kinetic equations are equivalent to the SBP system investigated in

All possible biochemical interactions that represent the ligand binding with Type I receptors and further recruitment of Type II receptors are:

The chemical interaction of Case II can easily be obtained from the interactions (_{1} to _{8}) of Case III (Figure _{±2} and _{±4} of Case III to zero. For the kinetics, we relied on the published data _{1}) is comparatively faster than the rate of BMPs and Type I receptors interactions _{1}_{2}) complex are not readily available, and hence, parameters were screened over the physiological ranges with values between [10^{-1 }^{1}] ^{-1 }^{-1} for the forward rates and [10^{-3 }^{0}] ^{-1} for the reverse reaction rates.

Simulation and discussion

To simulate the networks (as shown in Figure _{1} for Case I and _{1}_{2} for Case II, Case III) in the extracellular region is considered so a direct comparison can be made for the coefficient of variation _{1} and B_{1}_{2}.

The coefficient of variation (Λ) for _{1}_{2} remains very close to the coefficient of variation of _{1} as shown in Figure _{1} and _{1}_{2} (as shown in Figure _{2} brings the coefficient of variation of _{1}_{2} into very close agreement with the coefficient of variation of _{1} as shown in Figure _{1}_{2} is approximately equal to that of _{1}.

Comparison of Λ

**Comparison of Λ**. **a) **The coefficient of variation of _{1} (calculated from Case I Figure 3)and _{1}_{2} complexes (calculated from Case II Figure 3) is compared. The variability of the system seems to be invariant in the presence of Type II. **b) **Concentration dependency of Λ as a function of _{2}. **c) **Same as plot "a", however, direct interaction of BMP and Type II is allowed as in Case III, Figure 3. It's clear that the stochasticity of the system does not change over the range of values tested. **d) **Summary of _{1}_{2} formation and its impact on signaling noise.

Additionally, it is also found from the simulated result that the rate at which the BMP:Type I recruits Type II receptor (Case II in Figure

Comparison between SSA and Direct SS method

**Comparison between SSA and Direct SS method**. **a) **In Gillespie's method larger 'End Time' (ET) is required (which translates into a higher processing cost and time) to ensure the accuracy of outcome. Three different ET: 280 **b) **Effect of kinetics associated with _{1} interacting with _{2}. The steps of interactions are clearly shown in Case II of Figure 3.

Benchmarking of Direct SS approximation method

Carrying out large-scale stochastic simulation can be time consuming but calculation of the approximate solution via a truncated state-space can greatly improve the speed. In order to show the performance improvement in terms of computational cost and time of direct SS approximation in the analysis of stochastic biochemical networks, we benchmarked the method by comparing it with Gillespie's stochastic simulation algorithm (SSA) method

Benchmarking of Direct SS approximation method

**Benchmarking of Direct SS approximation method**. Benchmarking of Direct SS approximation method.

The processing time and computed Λ values for a target _{1}_{2} = 20 for Case II, Figure

Benchmarking: Gillespie's SSA and Direct SS approximation for a target _{1}_{2} = 20

**Method**

**End Time (ET) in Gillespie's SSA(hrs)**

**Processing Time (sec)**

**Λ**

Direct SS approximation

Not Applicable

0.4 -0.6

0.1707

Gillespie's SSA

28000

90-95

0.1705

2800

8-10

0.1878

1390

4-5

0.2254

In Table

Conclusions

In this study, we illustrate an approach of determining the steady state probability distribution efficiently to carry on continuation in multiple variables within a large-scale parameter screen. The approach is demonstrated further with a couple of applications, where we investigated 1) the dynamic dependency of a class of proteins, known as SBPs, in the regulation of BMP signaling, and 2) the binding of Type II receptor in BMP signaling. The results suggest that the recruitment of a type II receptor in BMP signaling doesn't affect the stochasticity of the system over the range of concentration and parameters investigated. Direct calculation of the SS probability distribution can be successfully applied to systems with a unique deterministic SS solution, and future work will investigate similar approaches for other biochemical systems.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MSK designed the research, contributed to algorithm development, performed simulation and data analysis, and wrote the manuscript. GB contributed to algorithm development, discussed the results and wrote the manuscript. DU contributed in research design, discussed the results and wrote the paper. All authors contributed to replying to reviewers' comments and approved the final manuscript of the paper.

Acknowledgements

Based on “Steady state probability approximation applied to stochastic model of biological network”, by Shahriar Karim, David M Umulis and Gregery T Buzzard which appeared in

We would like to thank Purdue University, West Lafayette, IN 47907, for financial support.

This article has been published as part of