Biomolecular Reaction Networks Research Articles

Structural identifiability of biomolecular controller motifs with and without flow measurements as model output.

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

The Internal Model Principle for Biomolecular Control Theory

The well-known Internal Model Principle (IMP) is a cornerstone of modern control theory. It stipulates the necessary conditions for asymptotic robustness of disturbance-prone dynamical systems by asserting that such a system must embed a subsystem in a feedback loop, and this subsystem must be able to reduplicate the dynamic disturbance using only the regulated variable as the input. The insights provided by IMP can help in both designing suitable controllers and also in analysing the regulatory mechanisms in complex systems. So far the application of IMP in biology has been case-specific and adhoc, primarily due to the lack of generic versions of the IMP for biomolecular reaction networks that model biological processes. In this short article we highlight the need for an IMP in biology and discuss a recently developed version of it for biomolecular networks that exhibit maximal Robust Perfect Adaptation (maxRPA) by being robust to the maximum number of disturbance sources.

DeepCME: A deep learning framework for computing solution statistics of the chemical master equation.

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov's forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the "policy function". This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.

Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

Cellular reactions have a multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary across orders of magnitude. This diversity naturally leads to hybrid models that combine continuous and discrete modeling regimes. In order to capture this multi-scale nature, we proposed jump-diffusion approximations in a previous study. The key idea was to partition reactions into fast and slow groups, and then to combine a Markov jump updating scheme for the slow group with a diffusion (Langevin) updating scheme for the fast group. In this study we show that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies a hybrid master equation that combines terms from the chemical master equation and from the Fokker–Planck equation. Inspired by the method of conditional moments, we propose a efficient method to solve this master equation using the moments of reaction counters of the fast reactions given the reaction counters of the slow reactions. For each time point of interest, we then solve a set of maximum entropy problems in order to recover the conditional probability density from its moments. This finally allows us to reconstruct the complete joint probability density over all reaction counters and hence obtain an approximate solution of the hybrid master equation. Finally, we show the accuracy of the method applied to a simple multi-scale conversion process.

Programming DNA-Based Biomolecular Reaction Networks on Cancer Cell Membranes.

DNA is a highly programmable biomolecule and has been used to construct biological circuits for different purposes. An important development of DNA circuits is to process the information on receptors on cell membranes. In this Communication, we introduce an architecture to program localized DNA-based biomolecular reaction networks on cancer cell membranes. Based on our architecture, various types of reaction networks have been experimentally demonstrated, from simple linear cascades to reaction networks of complex structures. These localized DNA-based reaction networks can be used for medical applications such as cancer cell detection. Compared to prior work on DNA circuits for evaluating cell membrane receptors, the DNA circuits made by our architecture have several major advantages including simpler design, lower leak, lower cost, and higher signal-to-background ratio.

The Effect of Loads in Molecular Communications

The ability of cells to sense and respond to their environment is encoded in biomolecular reaction networks, in which information travels through processes such as production, modification, and removal of biomolecules. Recent advances in biotechnology have made it possible to reengineer these physical processes to the point where synthetic biomolecular circuits can be inserted into cells to program cell behavior for useful functionalities. These circuits are often designed in a bottom-up fashion with smaller components connected to form complex systems. In a bottom-up approach to design, it is highly desirable that circuit components behave modularly, that is, the input-output behavior of a module characterized in isolation remains unchanged when the context changes. Unfortunately, due to the physical processes by which information is communicated from one biomolecular circuit module to the other, the lack of modularity is often a problem. In fact, the input-output behavior of a module depends on both direct connectivity to other modules, due to loading effects, and indirect connectivity arising from loads applied to shared cellular resources. In this paper, we summarize the published work illustrating how the means of molecular communication lead to these problems. Specifically, we review the concept of retroactivity, which has been proposed to capture loading problems within a “signals and systems” framework, allowing for engineering solutions that restore modularity.

Using rxncon to Develop Rule-Based Models.

We present a protocol for building, validating, and simulating models of signal transduction networks. These networks are challenging modeling targets due to the combinatorial complexity and sparse data, which have made it a major challenge even to formalize the current knowledge. To address this, the community has developed methods to model biomolecular reaction networks based on site dynamics. The strength of this approach is that reactions and states can be defined at variable resolution, which makes it possible to adapt the model resolution to the empirical data. This improves both scalability and accuracy, making it possible to formalize large models of signal transduction networks. Here, we present a method to build and validate large models of signal transduction networks. The workflow is based on rxncon, the reaction-contingency language. In a five-step process, we create a mechanistic network model, convert it into an executable Boolean model, use the Boolean model to evaluate and improve the network, and finally export the rxncon model into a rule-based format. We provide an introduction to the rxncon language and an annotated, step-by-step protocol for the workflow. Finally, we create a small model of the insulin signaling pathway to illustrate the protocol, together with some of the challenges-and some of their solutions-in modeling signal transduction.

Modular DNA strand-displacement controllers for directing material expansion

Soft materials that swell or change shape in response to external stimuli show extensive promise in regenerative medicine, targeted therapeutics, and soft robotics. Generally, a stimulus for shape change must interact directly with the material, limiting the types of stimuli that may be used and necessitating high stimulus concentrations. Here, we show how DNA strand-displacement controllers within hydrogels can mediate size change by interpreting, amplifying, and integrating stimuli and releasing signals that direct the response. These controllers tune the time scale and degree of DNA-crosslinked hydrogel swelling and can actuate dramatic material size change in response to <100 nM of a specific biomolecular input. Controllers can also direct swelling in response to small molecules or perform logic. The integration of these stimuli-responsive materials with biomolecular circuits is a major step towards autonomous soft robotic systems in which sensing and actuation are implemented by biomolecular reaction networks.

Antithetic proportional-integral feedback for reduced variance and improved control performance of stochastic reaction networks.

The ability of a cell to regulate and adapt its internal state in response to unpredictable environmental changes is called homeostasis and this ability is crucial for the cell's survival and proper functioning. Understanding how cells can achieve homeostasis, despite the intrinsic noise or randomness in their dynamics, is fundamentally important for both systems and synthetic biology. In this context, a significant development is the proposed antithetic integral feedback (AIF) motif, which is found in natural systems, and is known to ensure robust perfect adaptation for the mean dynamics of a given molecular species involved in a complex stochastic biomolecular reaction network. From the standpoint of applications, one drawback of this motif is that it often leads to an increased cell-to-cell heterogeneity or variance when compared to a constitutive (i.e. open-loop) control strategy. Our goal in this paper is to show that this performance deterioration can be countered by combining the AIF motif and a negative feedback strategy. Using a tailored moment closure method, we derive approximate expressions for the stationary variance for the controlled network that demonstrate that increasing the strength of the negative feedback can indeed decrease the variance, sometimes even below its constitutive level. Numerical results verify the accuracy of these results and we illustrate them by considering three biomolecular networks with two types of negative feedback strategies. Our computational analysis indicates that there is a trade-off between the speed of the settling-time of the mean trajectories and the stationary variance of the controlled species; i.e. smaller variance is associated with larger settling-time.

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks.

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.

Energy-based analysis of biomolecular pathways.

Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work, we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein [sodium-glucose transport protein 1 (SGLT1)]. The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.

Reconstructing dynamic molecular states from single-cell time series.

The notion of state for a system is prevalent in the quantitative sciences and refers to the minimal system summary sufficient to describe the time evolution of the system in a self-consistent manner. This is a prerequisite for a principled understanding of the inner workings of a system. Owing to the complexity of intracellular processes, experimental techniques that can retrieve a sufficient summary are beyond our reach. For the case of stochastic biomolecular reaction networks, we show how to convert the partial state information accessible by experimental techniques into a full system state using mathematical analysis together with a computational model. This is intimately related to the notion of conditional Markov processes and we introduce the posterior master equation and derive novel approximations to the corresponding infinite-dimensional posterior moment dynamics. We exemplify this state reconstruction approach using both in silico data and single-cell data from two gene expression systems in Saccharomyces cerevisiae, where we reconstruct the dynamic promoter and mRNA states from noisy protein abundance measurements.

Python to accelerate embedded SoC design

We present SysPy (System Python) a tool which exploits the strengths of the popular Python scripting language to boost design productivity of embedded System on Chips for FPGAs. SysPy acts as a “glue” software between mature HDLs, ready-to-use VHDL components and programmable processor soft IP cores. SysPy can be used to: (i) automatically translate hardware components described in Python into synthesizable VHDL, (ii) capture top-level structural descriptions of processor-centric SoCs in Python, (iii) implement all the steps necessary to compile the user's C code for an instruction set processor core and generate processor specific Tcl scripts that import to the design project all the necessary HDL files of the processor's description and instantiate/connect the core to other blocks in a synthesizable top-level Python description. Moreover, we have developed a Hardware Abstraction Layer (HAL) in Python which allows user applications running in a host PC to utilize effortlessly the SoC's resources in the FPGA. SysPy's design capabilities, when complemented with the developed HAL software API, provide all the necessary tools for hw/sw partitioning and iterative design for efficient SoC's performance tuning. We demonstrate how SysPy's design flow and functionalities can be used by building a processor-centric embedded SoC for computational systems biology. The designed SoC, implemented using a Xilinx Virtex-5 FPGA, combines the flexibility of a programmable soft processor core (Leon3) with the high performance of an application specific core to simulate flexibly and efficiently the stochastic behavior of large size biomolecular reaction networks. Such networks are essential for studying the dynamics of complex biological systems consisting of multiple interacting pathways.

Optimizing personalized treatment of oral mucositis secondary to cancer therapy through systems biology.

e19690 Background: Oral mucositis (OM) can result in clinically significant adverse outcomes that require high resource utilization. Clinical trials for mucositis drug development have typically incorporated a cohort-based methodology. However, patients vary considerably in risk and severity of OM. In addition, the modeling for OM pathobiology is complex and includes up-regulation of NF-kB and TNF-a. We therefore tested the hypothesis that an individual patient-oriented systems biology approach to mucositis management will maximize treatment efficacy, while reducing drug dosage. Methods: A mathematical model of the biomolecular reaction network describing the dynamics of NF-kB and TNF-a up-regulation was developed using data derived from published studies. To represent inter-subject variation, 10,000 in silico subjects were generated, with the reaction rate constants of each being randomly distributed by up to ± 30% relative to a pre-specified base-case. OM treatment was optimized using a genetic algorithm designed to down-regulate TNF-a levels to a pre-determined level using the minimum possible drug dosage. Two in silico subjects who deviated from average cohort behavior were randomly selected for comparison. Results: When optimized for the individual, required drug dosage for the first in silico subject was 20% of the cohort-optimized dosage on average. However, TNF-a levels for the individual were maintained within 5% of the average case. The optimal drug treatment for the second subject consisted of the maximum allowable drug dosage, representing a 10-fold increase versus the cohort-optimized case. Even with maximum drug dosage, TNF-a levels were at least 7.5-fold higher than the average case. In all cases, NF-kB dynamic behavior reflected that seen for TNF-a. Conclusions: Individually-optimized OM treatment may improve therapeutic efficacy in selected patients, while limiting drug dosage and associated side-effects. Supported by NIH Career Development Award K23DE016946 and NIH grant P41RR013186.

A Computational Tool for Monte Carlo Simulations of Biomolecular Reaction Networks Modeled on Physical Principles

Deciphering and designing complex biomolecular networks in the cell are the goals of systems and synthetic biology, respectively. The effects of localization, spatial heterogeneity, and molecular fluctuations in biomolecular networks are not well understood. We present a theoretical approach based on physical principles to accurately simulate biomolecular networks using the Monte Carlo method. Incorporating this theory, a computational tool named Monte Carlo biomolecular simulator (MBS) was developed, enabling studies of biomolecular kinetics with both spatial and temporal resolutions. The accuracy of MBS was verified by comparison against the classical deterministic approaches. Furthermore, the effects of localization, spatial heterogeneity, and molecular fluctuations were studied in three simulated model systems, showing their impact on the overall reaction kinetics. This work demonstrates the unique insights that can be discovered by considering the subtle effects that can be created by the spatial and temporal kinetics of biomolecular reaction networks.

Stochastic simulation and analysis of biomolecular reaction networks

BackgroundIn recent years, several stochastic simulation algorithms have been developed to generate Monte Carlo trajectories that describe the time evolution of the behavior of biomolecular reaction networks. However, the effects of various stochastic simulation and data analysis conditions on the observed dynamics of complex biomolecular reaction networks have not recieved much attention. In order to investigate these issues, we employed a a software package developed in out group, called Biomolecular Network Simulator (BNS), to simulate and analyze the behavior of such systems. The behavior of a hypothetical two gene in vitro transcription-translation reaction network is investigated using the Gillespie exact stochastic algorithm to illustrate some of the factors that influence the analysis and interpretation of these data.ResultsSpecific issues affecting the analysis and interpretation of simulation data are investigated, including: (1) the effect of time interval on data presentation and time-weighted averaging of molecule numbers, (2) effect of time averaging interval on reaction rate analysis, (3) effect of number of simulations on precision of model predictions, and (4) implications of stochastic simulations on optimization procedures.ConclusionThe two main factors affecting the analysis of stochastic simulations are: (1) the selection of time intervals to compute or average state variables and (2) the number of simulations generated to evaluate the system behavior.

Python Unleashed on Systems Biology

Researchers at Cornell University have built an open source software system to model biomolecular reaction networks. SloppyCell is written in Python and uses third-party libraries extensively, but it also does some fun things with on-the-fly code generation and parallel programming.