The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade

Low-dimensional projection of stochastic cell-signalling dynamics via a variational approach

Noise and fluctuations play vital roles in signal transduction in cells. Various numerical techniques for its simulation have been proposed, most of which are not efficient in cellular networks with a wide spectrum of timescales. In this paper, based on a recently developed variational technique, low-dimensional structures embedded in complex stochastic reaction dynamics are unfolded which sheds light on new design principles of efficient simulation algorithm for treating noise in the mesoscopic world. This idea is effectively demonstrated in several popular regulation models with an empirical selection of test functions according to their reaction geometry, which not only captures complex distribution profiles of different molecular species but also considerably speeds up the computation.

Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system

Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.

A multi-scale feedback control system model for wound healing electrical activity: therapeutic device/protocol implications

Regulation, growth and healing in biological systems involve many interconnected and interdependent processes that include chemical and electrical mechanisms of action. Unfortunately, the significant contributions that electrical events provide are often overlooked; resulting in a poor transfer of knowledge from science, to engineering and finally to therapy. Wound site electrical processes can influence cell migration, fluid transport, cellular signaling events, gene expression, cell differentiation and cell proliferation; affecting both form and function at the cell, tissue and organ levels. Wound healing, and its interrelationships with transport, regeneration, and growth, cannot be understood or therapeutically assisted unless both chemical and electrical activities associated with the healing process are addressed.

An effective method for computing the noise in biochemical networks

We present a simple yet effective method, which is based on power series expansion, for computing exact binomial moments that can be in turn used to compute steady-state probability distributions as well as the noise in linear or nonlinear biochemical reaction networks. When the method is applied to representative reaction networks such as the ON-OFF models of gene expression, gene models of promoter progression, gene auto-regulatory models, and common signaling motifs, the exact formulae for computing the intensities of noise in the species of interest or steady-state distributions are analytically given. Interestingly, we find that positive (negative) feedback does not enlarge (reduce) noise as claimed in previous works but has a counter-intuitive effect and that the multi-OFF (or ON) mechanism always attenuates the noise in contrast to the common ON-OFF mechanism and can modulate the noise to the lowest level independently of the mRNA mean. Except for its power in deriving analytical expressions for distributions and noise, our method is programmable and has apparent advantages in reducing computational cost.

Theory of active transport in filopodia and stereocilia

The biological processes in elongated organelles of living cells are often regulated by molecular motor transport. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean-field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean-field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers toward the tip, even the concentration bump of G actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G actin along the filopodium is rather nontrivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this nonmonotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

Noise propagation in two-step series MAPK cascade

Series MAPK enzymatic cascades, ubiquitously found in signaling networks, act as signal amplifiers and play a key role in processing information during signal transduction in cells. In activated cascades, cell-to-cell variability or noise is bound to occur and thereby strongly affects the cellular response. Commonly used linearization method (LM) applied to Langevin type stochastic model of the MAPK cascade fails to accurately predict intrinsic noise propagation in the cascade. We prove this by using extensive stochastic simulations for various ranges of biochemical parameters. This failure is due to the fact that the LM ignores the nonlinear effects on the noise. However, LM provides a good estimate of the extrinsic noise propagation. We show that the correct estimate of intrinsic noise propagation in signaling networks that contain at least one enzymatic step can be obtained only through stochastic simulations. Noise propagation in the cascade depends on the underlying biochemical parameters which are often unavailable. Based on a combination of global sensitivity analysis (GSA) and stochastic simulations, we developed a systematic methodology to characterize noise propagation in the cascade. GSA predicts that noise propagation in MAPK cascade is sensitive to the total number of upstream enzyme molecules and the total number of molecules of the two substrates involved in the cascade. We argue that the general systematic approach proposed and demonstrated on MAPK cascade must accompany noise propagation studies in biological networks.

How does the antagonism between capping and anti-capping proteins affect actin network dynamics?

Actin-based cell motility is essential to many biological processes. We built a simplified, three-dimensional computational model and subsequently performed stochastic simulations to study the growth dynamics of lamellipodia-like branched networks. In this work, we shed light on the antagonism between capping and anti-capping proteins in regulating actin dynamics in the filamentous network. We discuss detailed mechanisms by which capping and anti-capping proteins affect the protrusion speed of the actin network and the rate of nucleation of filaments. We computed a phase diagram showing the regimes of motility enhancement and inhibition by these proteins. Our work shows that the effects of capping and anti-capping proteins are mainly transmitted by modulation of the filamentous network density and local availability of monomeric actin. We discovered that the combination of the capping/anti-capping regulatory network with nucleation-promoting proteins introduces robustness and redundancy in cell motility machinery, allowing the cell to easily achieve maximal protrusion speeds under a broader set of conditions. Finally, we discuss distributions of filament lengths under various conditions and speculate on their potential implication for the emergence of filopodia from the lamellipodial network.

Stochastic approaches in systems biology

The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches--the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker-Planck equation and the two-moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts.

Mechano-chemical feedbacks regulate actin mesh growth in lamellipodial protrusions

During cell motion on a substratum, eukaryotic cells project sheetlike lamellipodia which contain a dynamically remodeling three-dimensional actin mesh. A number of regulatory proteins and subtle mechano-chemical couplings determine the lamellipodial protrusion dynamics. To study these processes, we constructed a microscopic physico-chemical computational model, which incorporates a number of fundamental reaction and diffusion processes, treated in a fully stochastic manner. Our work sheds light on the way lamellipodial protrusion dynamics is affected by the concentrations of actin and actin-binding proteins. In particular, we found that protrusion speed saturates at very high actin concentrations, where filament nucleation does not keep up with protrusion. This results in sparse filamentous networks, and, consequently, high resistance forces on individual filaments. We also observed maxima in lamellipodial growth rates as a function of Arp2/3, a nucleating protein, and capping proteins. We provide detailed physical explanations behind these effects. In particular, our work supports the actin-funneling-hypothesis explanation of protrusion speed enhancement at low capping protein concentrations. Our computational results are in agreement with a number of related experiments. Overall, our work emphasizes that elongation and nucleation processes work highly cooperatively in determining the optimal protrusion speed for the actin mesh in lamellipodia.

Design of active transport must be highly intricate: a possible role of myosin and Ena/VASP for G-actin transport in filopodia

Recent modeling of filopodia--the actin-based cell organelles employed for sensing and motility--reveals that one of the key limiting factors of filopodial length is diffusional transport of G-actin monomers to the polymerizing barbed ends. We have explored the possibility of active transport of G-actin by myosin motors, which would be an expected biological response to overcome the limitation of a diffusion-based process. We found that in a straightforward implementation of active transport the increase in length was unimpressive, < or = 30%, due to sequestering of G-actin by freely diffusing motors. However, artificially removing motor sequestration reactions led to approximately threefold increases in filopodial length, with the transport being mainly limited by the motors failing to detach from the filaments near the tip, clogging the cooperative conveyer belt dynamics. Making motors sterically transparent led to a qualitative change of the dynamics to a different regime of steady growth without a stationary length. Having identified sequestration and clogging as ubiquitous constraints to motor-driven transport, we devised and tested a speculative means to sidestep these limitations in filopodia by employing cross-linking and putative scaffolding roles of Ena/VASP proteins. We conclude that a naïve design of molecular-motor-based active transport would almost always be inefficient--an intricately organized kinetic scheme, with finely tuned rate constants, is required to achieve high-flux transport.

A critical quantity for noise attenuation in feedback systems

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as "signed activation time". We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.

Spectral solutions to stochastic models of gene expression with bursts and regulation

Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.

Molecular noise of capping protein binding induces macroscopic instability in filopodial dynamics

Capping proteins are among the most important regulatory proteins involved in controlling complicated stochastic dynamics of filopodia, which are dynamic finger-like protrusions used by eukaryotic motile cells to probe their environment and help guide cell motility. They attach to the barbed end of a filament and prevent polymerization, leading to effective filament retraction due to retrograde flow. When we simulated filopodial growth in the presence of capping proteins, qualitatively different dynamics emerged, compared with actin-only system. We discovered that molecular noise due to capping protein binding and unbinding leads to macroscopic filopodial length fluctuations, compared with minuscule fluctuations in the actin-only system. Thus, our work shows that molecular noise of signaling proteins may induce micrometer-scale growth-retraction cycles in filopodia. When capped, some filaments eventually retract all the way down to the filopodial base and disappear. This process endows filopodium with a finite lifetime. Additionally, the filopodia transiently grow several times longer than in actin-only system, since less actin transport is required because of bundle thinning. We have also developed an accurate mean-field model that provides qualitative explanations of our numerical simulation results. Our results are broadly consistent with experiments, in terms of predicting filopodial growth retraction cycles and the average filopodial lifetimes.

Investigating the two-moment characterisation of subcellular biochemical networks

While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.

A stochastic spectral analysis of transcriptional regulatory cascades

The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here, we introduce the spectral method for computation of the joint probability distribution over all species in a biological network. The spectral method exploits the natural eigenfunctions of the master equation of birth-death processes to solve for the joint distribution of modules within the network, which then inform each other and facilitate calculation of the entire joint distribution. We illustrate the method on a ubiquitous case in nature: linear regulatory cascades. The efficiency of the method makes possible numerical optimization of the input and regulatory parameters, revealing design properties of, e.g., the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of up-regulations outperforms a chain of down-regulations. We anticipate that this numerical approach may be useful for modeling noise in a variety of small network topologies in biology.

Elimination of fast variables in chemical Langevin equations

Internal and external fluctuations are ubiquitous in cellular signaling processes. Because biochemical reactions often evolve on disparate time scales, mathematical perturbation techniques can be invoked to reduce the complexity of stochastic models. Previous work in this area has focused on direct treatment of the master equation. However, eliminating fast variables in the chemical Langevin equation is also an important problem. We show how to solve this problem by utilizing a partial equilibrium assumption. Our technique is applied to a simple birth-death-dimerization process and a more involved gene regulation network, demonstrating great computational efficiency. Excellent agreement is found with results computed from exact stochastic simulations. We compare our approach with existing reduction schemes and discuss avenues for future improvement.

The stochastic dynamics of filopodial growth

A filopodium is a cytoplasmic projection, exquisitely built and regulated, which extends from the leading edge of the migrating cell, exploring the cell's neighborhood. Commonly, filopodia grow and retract after their initiation, exhibiting rich dynamical behaviors. We model the growth of a filopodium based on a stochastic description which incorporates mechanical, physical, and biochemical components. Our model provides a full stochastic treatment of the actin monomer diffusion and polymerization of each individual actin filament under stress of the fluctuating membrane. We investigated the length distribution of individual filaments in a growing filopodium and studied how it depends on various physical parameters. The distribution of filament lengths turned out to be narrow, which we explained by the negative feedback created by the membrane load and monomeric G-actin gradient. We also discovered that filopodial growth is strongly diminished upon increasing retrograde flow, suggesting that regulating the retrograde flow rate would be a highly efficient way to control filopodial extension dynamics. The filopodial length increases as the membrane fluctuations decrease, which we attributed to the unequal loading of the membrane force among individual filaments, which, in turn, results in larger average polymerization rates. We also observed significant diffusional noise of G-actin monomers, which leads to smaller G-actin flux along the filopodial tube compared with the prediction using the diffusion equation. Overall, partial cancellation of these two fluctuation effects allows a simple mean field model to rationalize most of our simulation results. However, fast fluctuations significantly renormalize the mean field model parameters. The biological significance of our filopodial model and avenues for future development are also discussed.

Evolution of complex probability distributions in enzyme cascades

Unusual probability distribution profiles, including transient multi-peak distributions, have been observed in computer simulations of cell signaling dynamics. The emergence of these complex distributions cannot be explained using either deterministic chemical kinetics or simple Gaussian noise approximation. To develop physical insights into the origin of complex distributions in stochastic cell signaling, we compared our approximate analytical solutions of signaling dynamics with the exact numerical simulations. Our results are based on studying signaling in 2-step and 3-step enzyme amplification cascades that are among the most common building blocks of cellular protein signaling networks. We have found that while the multi-peak distributions are typically transient, and eventually evolve into single peak distributions, in certain cases these distributions may be stable in the limit of long times. We also have shown that introducing positive feedback loops results in diminution of the probability distribution complexity.

Stochastic resonant signaling in enzyme cascades

We observe the phenomenon of stochastic resonant signaling in signal amplification enzyme cascades, where certain optimal reaction rates minimize the average threshold-crossing time. We develop a new analytical technique to obtain the mean first passage time, based on a novel decomposition of the master equation. Our analytical results are in good agreement with the exact numerical simulations. We demonstrate that resonant behavior may be a ubiquitous phenomenon in stochastic threshold crossing in cell signaling. The physical principles behind this phenomenon are elucidated.