Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models

Benchmarking methods for computing local sensitivities in ordinary differential equation models at dynamic and steady states

Estimating parameters of dynamic models from experimental data is a challenging, and often computationally-demanding task. It requires a large number of model simulations and objective function gradient computations, if gradient-based optimization is used. In many cases, steady-state computation is a part of model simulation, either due to steady-state data or an assumption that the system is at steady state at the initial time point. Various methods are available for steady-state and gradient computation. Yet, the most efficient pair of methods (one for steady states, one for gradients) for a particular model is often not clear. In order to facilitate the selection of methods, we explore six method pairs for computing the steady state and sensitivities at steady state using six real-world problems. The method pairs involve numerical integration or Newton's method to compute the steady-state, and-for both forward and adjoint sensitivity analysis-numerical integration or a tailored method to compute the sensitivities at steady-state. Our evaluation shows that all method pairs provide accurate steady-state and gradient values, and that the two method pairs that combine numerical integration for the steady-state with a tailored method for the sensitivities at steady-state were the most robust, and amongst the most computationally-efficient. We also observed that while Newton's method for steady-state computation yields a substantial speedup compared to numerical integration, it may lead to a large number of simulation failures. Overall, our study provides a concise overview across current methods for computing sensitivities at steady state. While our study shows that there is no universally-best method pair, it also provides guidance to modelers in choosing the right methods for a problem at hand.

Kinetic Diagram Analysis: A Python Library for Calculating Steady-State Observables of Biochemical Systems Analytically

Kinetic diagrams are commonly used to represent biochemical systems in order to study phenomena such as free energy transduction and ion selectivity. While numerical methods are commonly used to analyze such kinetic networks, the diagram method by King, Altman and Hill makes it possible to construct exact algebraic expressions for steady-state observables in terms of the rate constants of the kinetic diagram. However, manually obtaining these expressions becomes infeasible for models of even modest complexity as the number of the required intermediate diagrams grows with the factorial of the number of states in the diagram. We developed Kinetic Diagram Analysis (KDA), a Python library that programmatically generates the relevant diagrams and expressions from a user-defined kinetic diagram. KDA outputs symbolic expressions for state probabilities and cycle fluxes at steady-state that can be symbolically manipulated and evaluated to quantify macroscopic system observables. We demonstrate the KDA approach for examples drawn from the biophysics of active secondary transmembrane transporters. For a generic 6-state antiporter model, we show how the introduction of a single leakage transition reduces transport efficiency by quantifying substrate turnover. We apply KDA to a real-world example, the 8-state free exchange model of the small multidrug resistance transporter EmrE of Hussey et al. (J. Gen. Physiol., 2020, 152, e201912437), where a change in transporter phenotype is achieved by biasing two different subsets of kinetic rates: alternating access and substrate unbinding rates. KDA is made available as open source software under the GNU General Public License version 3.

Patient-specific computational models predict prognosis in B cell lymphoma by quantifying pro-proliferative and anti-apoptotic signatures from genetic sequencing data

Genetic heterogeneity and co-occurring driver mutations impact clinical outcomes in blood cancers, but predicting the emergent effect of co-occurring mutations that impact multiple complex and interacting signalling networks is challenging. Here, we used mathematical models to predict the impact of co-occurring mutations on cellular signalling and cell fates in diffuse large B cell lymphoma and multiple myeloma. Simulations predicted adverse impact on clinical prognosis when combinations of mutations induced both anti-apoptotic (AA) and pro-proliferative (PP) signalling. We integrated patient-specific mutational profiles into personalised lymphoma models, and identified patients characterised by simultaneous upregulation of anti-apoptotic and pro-proliferative (AAPP) signalling in all genomic and cell-of-origin classifications (8-25% of patients). In a discovery cohort and two validation cohorts, patients with upregulation of neither, one (AA or PP), or both (AAPP) signalling states had good, intermediate and poor prognosis respectively. Combining AAPP signalling with genetic or clinical prognostic predictors reliably stratified patients into striking prognostic categories. AAPP patients in poor prognosis genetic clusters had 7.8 months median overall survival, while patients lacking both features had 90% overall survival at 120 months in a validation cohort. Personalised computational models enable identification of novel risk-stratified patient subgroups, providing a valuable tool for future risk-adapted clinical trials.

Control Theory and Systems Biology: Potential Applications in Neurodegeneration and Search for Therapeutic Targets

Control theory, a well-established discipline in engineering and mathematics, has found novel applications in systems biology. This interdisciplinary approach leverages the principles of feedback control and regulation to gain insights into the complex dynamics of cellular and molecular networks underlying chronic diseases, including neurodegeneration. By modeling and analyzing these intricate systems, control theory provides a framework to understand the pathophysiology and identify potential therapeutic targets. Therefore, this review examines the most widely used control methods in conjunction with genomic-scale metabolic models in the steady state of the multi-omics type. According to our research, this approach involves integrating experimental data, mathematical modeling, and computational analyses to simulate and control complex biological systems. In this review, we find that the most significant application of this methodology is associated with cancer, leaving a lack of knowledge in neurodegenerative models. However, this methodology, mainly associated with the Minimal Dominant Set (MDS), has provided a starting point for identifying therapeutic targets for drug development and personalized treatment strategies, paving the way for more effective therapies.

Computational modeling of DLBCL predicts response to BH3-mimetics

In healthy cells, pro- and anti-apoptotic BCL2 family and BH3-only proteins are expressed in a delicate equilibrium. In contrast, this homeostasis is frequently perturbed in cancer cells due to the overexpression of anti-apoptotic BCL2 family proteins. Variability in the expression and sequestration of these proteins in Diffuse Large B cell Lymphoma (DLBCL) likely contributes to variability in response to BH3-mimetics. Successful deployment of BH3-mimetics in DLBCL requires reliable predictions of which lymphoma cells will respond. Here we show that a computational systems biology approach enables accurate prediction of the sensitivity of DLBCL cells to BH3-mimetics. We found that fractional killing of DLBCL, can be explained by cell-to-cell variability in the molecular abundances of signaling proteins. Importantly, by combining protein interaction data with a knowledge of genetic lesions in DLBCL cells, our in silico models accurately predict in vitro response to BH3-mimetics. Furthermore, through virtual DLBCL cells we predict synergistic combinations of BH3-mimetics, which we then experimentally validated. These results show that computational systems biology models of apoptotic signaling, when constrained by experimental data, can facilitate the rational assignment of efficacious targeted inhibitors in B cell malignancies, paving the way for development of more personalized approaches to treatment.

Efficient computation of adjoint sensitivities at steady-state in ODE models of biochemical reaction networks

Dynamical models in the form of systems of ordinary differential equations have become a standard tool in systems biology. Many parameters of such models are usually unknown and have to be inferred from experimental data. Gradient-based optimization has proven to be effective for parameter estimation. However, computing gradients becomes increasingly costly for larger models, which are required for capturing the complex interactions of multiple biochemical pathways. Adjoint sensitivity analysis has been pivotal for working with such large models, but methods tailored for steady-state data are currently not available. We propose a new adjoint method for computing gradients, which is applicable if the experimental data include steady-state measurements. The method is based on a reformulation of the backward integration problem to a system of linear algebraic equations. The evaluation of the proposed method using real-world problems shows a speedup of total simulation time by a factor of up to 4.4. Our results demonstrate that the proposed approach can achieve a substantial improvement in computation time, in particular for large-scale models, where computational efficiency is critical.

Crosstalk and ultrasensitivity in protein degradation pathways

Protein turnover is vital to cellular homeostasis. Many proteins are degraded efficiently only after they have been post-translationally “tagged” with a polyubiquitin chain. Ubiquitylation is a form of Post-Translational Modification (PTM): addition of a ubiquitin to the chain is catalyzed by E3 ligases, and removal of ubiquitin is catalyzed by a De-UBiquitylating enzyme (DUB). Nearly four decades ago, Goldbeter and Koshland discovered that reversible PTM cycles function like on-off switches when the substrates are at saturating concentrations. Although this finding has had profound implications for the understanding of switch-like behavior in biochemical networks, the general behavior of PTM cycles subject to synthesis and degradation has not been studied. Using a mathematical modeling approach, we found that simply introducing protein turnover to a standard modification cycle has profound effects, including significantly reducing the switch-like nature of the response. Our findings suggest that many classic results on PTM cycles may not hold in vivo where protein turnover is ubiquitous. We also found that proteins sharing an E3 ligase can have closely related changes in their expression levels. These results imply that it may be difficult to interpret experimental results obtained from either overexpressing or knocking down protein levels, since changes in protein expression can be coupled via E3 ligase crosstalk. Understanding crosstalk and competition for E3 ligases will be key in ultimately developing a global picture of protein homeostasis.

Previous work has shown that substrates of Post-Translational Modification (PTM) cycles can have coupled responses if those substrates share enzymes. This implies that modifications leading to substrate degradation (e.g. ubiquitylation by an E3 ligase) could introduce coupling in concentrations of substrates sharing a ligase. Using mathematical models, we found adding protein turnover to a PTM cycle diminishes both sensitivity and ultrasensitivity, particularly in models admitting long ubiquitin chains. We also found that proteins sharing an E3 ligase can indeed have coupled changes in both expression and sensitivity to signals. These results imply that accounting for crosstalk in protein degradation networks is crucial for the interpretation of results from a wide variety of common experimental perturbations to living systems.

What Will B Will B: Identifying Molecular Determinants of Diverse B-Cell Fate Decisions Through Systems Biology

B-cells are the poster child for cellular diversity and heterogeneity. The diverse repertoire of B lymphocytes, each expressing unique antigen receptors, provides broad protection against pathogens. However, B-cell diversity goes beyond unique antigen receptors. Side-stepping B-cell receptor (BCR) diversity through BCR-independent stimuli or engineered organisms with monoclonal BCRs still results in seemingly identical B-cells reaching a wide variety of fates in response to the same challenge. Identifying to what extent the molecular state of a B-cell determines its fate is key to gaining a predictive understanding of B-cells and consequently the ability to control them with targeted therapies. Signals received by B-cells through transmembrane receptors converge on intracellular molecular signaling networks, which control whether each B-cell divides, dies, or differentiates into a number of antibody-secreting distinct B-cell subtypes. The signaling networks that interpret these signals are well known to be susceptible to molecular variability and noise, providing a potential source of diversity in cell fate decisions. Iterative mathematical modeling and experimental studies have provided quantitative insight into how B-cells achieve distinct fates in response to pathogenic stimuli. Here, we review how systems biology modeling of B-cells, and the molecular signaling networks controlling their fates, is revealing the key determinants of cell-to-cell variability in B-cell destiny.

An algebraic method to calculate parameter regions for constrained steady-state distribution in stochastic reaction networks

Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuations. Although several methods have been proposed to detect and analyze the stability of steady states for deterministic models, these methods cannot be applied to stochastic reaction networks. In this paper, we propose an algorithm based on algebraic computations to calculate parameter regions for constrained steady-state distribution of stochastic reaction networks, in which the means and variances satisfy some given inequality constraints. To evaluate our proposed method, we perform computer simulations for three typical chemical reactions and demonstrate that the results obtained with our method are consistent with the simulation results.

Nongenetic origins of cell-to-cell variability in B lymphocyte proliferation

Rapid antibody production in response to invading pathogens requires the dramatic expansion of pathogen-derived antigen-specific B lymphocyte populations. Whether B cell population dynamics are based on stochastic competition between competing cell fates, as in the development of competence by the bacterium Bacillus subtilis, or on deterministic cell fate decisions that execute a predictable program, as during the development of the worm Caenorhabditis elegans, remains unclear. Here, we developed long-term live-cell microscopy of B cell population expansion and multiscale mechanistic computational modeling to characterize the role of molecular noise in determining phenotype heterogeneity. We show that the cell lineage trees underlying B cell population dynamics are mediated by a largely predictable decision-making process where the heterogeneity of cell proliferation and death decisions at any given timepoint largely derives from nongenetic heterogeneity in the founder cells. This means that contrary to previous models, only a minority of genetically identical founder cells contribute the majority to the population response. We computationally predict and experimentally confirm nongenetic molecular determinants that are predictive of founder cells' proliferative capacity. While founder cell heterogeneity may arise from different exposure histories, we show that it may also be due to the gradual accumulation of small amounts of intrinsic noise during the lineage differentiation process of hematopoietic stem cells to mature B cells. Our finding of the largely deterministic nature of B lymphocyte responses may provide opportunities for diagnostic and therapeutic development.

Petri Net computational modelling of Langerhans cell Interferon Regulatory Factor Network predicts their role in T cell activation

Langerhans cells (LCs) are able to orchestrate adaptive immune responses in the skin by interpreting the microenvironmental context in which they encounter foreign substances, but the regulatory basis for this has not been established. Utilising systems immunology approaches combining in silico modelling of a reconstructed gene regulatory network (GRN) with in vitro validation of the predictions, we sought to determine the mechanisms of regulation of immune responses in human primary LCs. The key role of Interferon regulatory factors (IRFs) as controllers of the human Langerhans cell response to epidermal cytokines was revealed by whole transcriptome analysis. Applying Boolean logic we assembled a Petri net-based model of the IRF-GRN which provides molecular pathway predictions for the induction of different transcriptional programmes in LCs. In silico simulations performed after model parameterisation with transcription factor expression values predicted that human LC activation of antigen-specific CD8 T cells would be differentially regulated by epidermal cytokine induction of specific IRF-controlled pathways. This was confirmed by in vitro measurement of IFN-γ production by activated T cells. As a proof of concept, this approach shows that stochastic modelling of a specific immune networks renders transcriptome data valuable for the prediction of functional outcomes of immune responses.

Tailored parameter optimization methods for ordinary differential equation models with steady-state constraints

BACKGROUND

Ordinary differential equation (ODE) models are widely used to describe (bio-)chemical and biological processes. To enhance the predictive power of these models, their unknown parameters are estimated from experimental data. These experimental data are mostly collected in perturbation experiments, in which the processes are pushed out of steady state by applying a stimulus. The information that the initial condition is a steady state of the unperturbed process provides valuable information, as it restricts the dynamics of the process and thereby the parameters. However, implementing steady-state constraints in the optimization often results in convergence problems.

RESULTS

In this manuscript, we propose two new methods for solving optimization problems with steady-state constraints. The first method exploits ideas from optimization algorithms on manifolds and introduces a retraction operator, essentially reducing the dimension of the optimization problem. The second method is based on the continuous analogue of the optimization problem. This continuous analogue is an ODE whose equilibrium points are the optima of the constrained optimization problem. This equivalence enables the use of adaptive numerical methods for solving optimization problems with steady-state constraints. Both methods are tailored to the problem structure and exploit the local geometry of the steady-state manifold and its stability properties. A parameterization of the steady-state manifold is not required. The efficiency and reliability of the proposed methods is evaluated using one toy example and two applications. The first application example uses published data while the second uses a novel dataset for Raf/MEK/ERK signaling. The proposed methods demonstrated better convergence properties than state-of-the-art methods employed in systems and computational biology. Furthermore, the average computation time per converged start is significantly lower. In addition to the theoretical results, the analysis of the dataset for Raf/MEK/ERK signaling provides novel biological insights regarding the existence of feedback regulation.

CONCLUSION

Many optimization problems considered in systems and computational biology are subject to steady-state constraints. While most optimization methods have convergence problems if these steady-state constraints are highly nonlinear, the methods presented recover the convergence properties of optimizers which can exploit an analytical expression for the parameter-dependent steady state. This renders them an excellent alternative to methods which are currently employed in systems and computational biology.

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

Distinct single-cell signaling characteristics are conferred by the MyD88 and TRIF pathways during TLR4 activation

Toll-like receptors (TLRs) recognize specific pathogen-associated molecular patterns and initiate innate immune responses through signaling pathways that depend on the adaptor proteins MyD88 (myeloid differentiation marker 88) or TRIF (TIR domain-containing adaptor protein-inducing interferon-β). TLR4, in particular, uses both adaptor proteins to activate the transcription factor nuclear factor κB (NF-κB); however, the specificity and redundancy of these two pathways remain to be elucidated. We developed a mathematical model to show how each pathway encodes distinct dynamical features of NF-κB activity and makes distinct contributions to the high variability observed in single-cell measurements. The assembly of a macromolecular signaling platform around MyD88 associated with receptors at the cell surface determined the timing of initial responses to generate a reliable, digital NF-κB signal. In contrast, ligand-induced receptor internalization into endosomes produced noisy, delayed, yet sustained NF-κB signals through TRIF. With iterative mathematical model development, we predicted the molecular mechanisms by which the MyD88- and TRIF-mediated pathways provide ligand concentration-dependent signaling dynamics that transmit information about the pathogen threat.

A multi-scale approach reveals that NF-κB cRel enforces a B-cell decision to divide

Understanding the functions of multi-cellular organs in terms of the molecular networks within each cell is an important step in the quest to predict phenotype from genotype. B-lymphocyte population dynamics, which are predictive of immune response and vaccine effectiveness, are determined by individual cells undergoing division or death seemingly stochastically. Based on tracking single-cell time-lapse trajectories of hundreds of B cells, single-cell transcriptome, and immunofluorescence analyses, we constructed an agent-based multi-modular computational model to simulate lymphocyte population dynamics in terms of the molecular networks that control NF-κB signaling, the cell cycle, and apoptosis. Combining modeling and experimentation, we found that NF-κB cRel enforces the execution of a cellular decision between mutually exclusive fates by promoting survival in growing cells. But as cRel deficiency causes growing B cells to die at similar rates to non-growing cells, our analysis reveals that the phenomenological decision model of wild-type cells is rooted in a biased race of cell fates. We show that a multi-scale modeling approach allows for the prediction of dynamic organ-level physiology in terms of intra-cellular molecular networks.