Rigorous Mapping of Data to Qualitative Properties of Parameter Values and Dynamics: A Case Study on a Two-Variable Lotka-Volterra System.

In this work, we describe mostly analytical work related to a novel approach to parameter identification for a two-variable Lotka-Volterra (LV) system. Specifically, this approach is qualitative, in that we aim not to determine precise values of model parameters but rather to establish relationships among these parameter values and properties of the trajectories that they generate, based on a small number of available data points. In this vein, we prove a variety of results about the existence, uniqueness, and signs of model parameters for which the trajectory of the system passes exactly through a set of three given data points, representing the smallest possible data set needed for identification of model parameter values. We find that in most situations such a data set determines these values uniquely; we also thoroughly investigate the alternative cases, which result in nonuniqueness or even nonexistence of model parameter values that fit the data. In addition to results about identifiability, our analysis provides information about the long-term dynamics of solutions of the LV system directly from the data without the necessity of estimating specific parameter values.

A data-driven model of the role of energy in sepsis.

Exposure to pathogens elicits a complex immune response involving multiple interdependent pathways. This response may mitigate detrimental effects and restore health but, if imbalanced, can lead to negative outcomes including sepsis. This complexity and need for balance pose a challenge for clinicians and have attracted attention from modelers seeking to apply computational tools to guide therapeutic approaches. In this work, we address a shortcoming of such past efforts by incorporating the dynamics of energy production and consumption into a computational model of the acute immune response. With this addition, we performed fits of model dynamics to data obtained from non-human primates exposed to Escherichia coli. Our analysis identifies parameters that may be crucial in determining survival outcomes and also highlights energy-related factors that modulate the immune response across baseline and altered glucose conditions.

Synchronization and locking in oscillators with flexible periods.

Entrainment of a nonlinear oscillator by a periodic external force is a much studied problem in nonlinear dynamics and characterized by the well-known Arnold tongues. The circle map is the simplest such system allowing for stable N:M entrainment where the oscillator produces N cycles for every M stimulus cycles. There are a number of experiments that suggest that entrainment to external stimuli can involve both a shift in the phase and an adjustment of the intrinsic period of the oscillator. Motivated by a recent model of Loehr et al. [J. Exp. Psychol.: Hum. Percept. Perform. 37, 1292 (2011)], we explore a two-dimensional map in which the phase and the period are allowed to update as a function of the phase of the stimulus. We characterize the number and stability of fixed points for different N:M-locking regions, specifically, 1:1, 1:2, 2:3, and their reciprocals, as a function of the sensitivities of the phase and period to the stimulus as well as the degree that the oscillator has a preferred period. We find that even in the limited number of locking regimes explored, there is a great deal of multi-stability of locking modes, and the basins of attraction can be complex and riddled. We also show that when the forcing period changes between a starting and final period, the rate of this change determines, in a complex way, the final locking pattern.

Mathematical modeling of energy consumption in the acute inflammatory response.

When a pathogen invades the body, an acute inflammatory response is activated to eliminate the intruder. In some patients, runaway activation of the immune system may lead to collateral tissue damage and, in the extreme, organ failure and death. Experimental studies have found an association between severe infections and depletion in levels of adenosine triphosphate (ATP), increase in nitric oxide production, and accumulation of lactate, suggesting that tissue energetics is compromised. In this work we present a differential equations model that incorporates the dynamics of ATP, nitric oxide, and lactate accompanying an acute inflammatory response and employ this model to explore their roles in shaping this response. The bifurcation diagram of the model system with respect to the pathogen growth rate reveals three equilibrium states characterizing the health, aseptic and septic conditions. We explore the domains of attraction of these states to inform the instantiation of heterogeneous virtual patient populations utilized in a survival analysis. We then apply the model to study alterations in the inflammatory response and survival outcomes in metabolically altered conditions such as hypoglycemia, hyperglycemia, and hypoxia.

Discrete Dynamical Modeling of Influenza Virus Infection Suggests Age-Dependent Differences in Immunity.

Immunosenescence, an age-related decline in immune function, is a major contributor to morbidity and mortality in the elderly. Older hosts exhibit a delayed onset of immunity and prolonged inflammation after an infection, leading to excess damage and a greater likelihood of death. Our study applies a rule-based model to infer which components of the immune response are most changed in an aged host. Two groups of BALB/c mice (aged 12 to 16 weeks and 72 to 76 weeks) were infected with 2 inocula: a survivable dose of 50 PFU and a lethal dose of 500 PFU. Data were measured at 10 points over 19 days in the sublethal case and at 6 points over 7 days in the lethal case, after which all mice had died. Data varied primarily in the onset of immunity, particularly the inflammatory response, which led to a 2-day delay in the clearance of the virus from older hosts in the sublethal cohort. We developed a Boolean model to describe the interactions between the virus and 21 immune components, including cells, chemokines, and cytokines, of innate and adaptive immunity. The model identifies distinct sets of rules for each age group by using Boolean operators to describe the complex series of interactions that activate and deactivate immune components. Our model accurately simulates the immune responses of mice of both ages and with both inocula included in the data (95% accurate for younger mice and 94% accurate for older mice) and shows distinct rule choices for the innate immunity arm of the model between younger and aging mice in response to influenza A virus infection.IMPORTANCE Influenza virus infection causes high morbidity and mortality rates every year, especially in the elderly. The elderly tend to have a delayed onset of many immune responses as well as prolonged inflammatory responses, leading to an overall weakened response to infection. Many of the details of immune mechanisms that change with age are currently not well understood. We present a rule-based model of the intrahost immune response to influenza virus infection. The model is fit to experimental data for young and old mice infected with influenza virus. We generated distinct sets of rules for each age group to capture the temporal differences seen in the immune responses of these mice. These rules describe a network of interactions leading to either clearance of the virus or death of the host, depending on the initial dosage of the virus. Our models clearly demonstrate differences in these two age groups, particularly in the innate immune responses.

APT-MCMC, a C++/Python implementation of Markov Chain Monte Carlo for parameter identification.

The inverse problem associated with fitting parameters of an ordinary differential equation (ODE) system to data is nonlinear and multimodal, which is of great challenge to gradient-based optimizers. Markov Chain Monte Carlo (MCMC) techniques provide an alternative approach to solving these problems and can escape local minima by design. APT-MCMC was created to allow users to setup ODE simulations in Python and run as compiled C++ code. It combines affine-invariant ensemble of samplers and parallel tempering MCMC techniques to improve the simulation efficiency. Simulations use Bayesian inference to provide probability distributions of parameters, which enable analysis of multiple minima and parameter correlation. Benchmark tests result in a 20×-60× speedup but 14% increase in memory usage against emcee, a similar MCMC package in Python. Several MCMC hyperparameters were analyzed: number of temperatures, ensemble size, step size, and swap attempt frequency. Heuristic tuning guidelines are provided for setting these hyperparameters.

DNA topology confers sequence specificity to nonspecific architectural proteins.

Topological constraints placed on short fragments of DNA change the disorder found in chain molecules randomly decorated by nonspecific, architectural proteins into tightly organized 3D structures. The bacterial heat-unstable (HU) protein builds up, counter to expectations, in greater quantities and at particular sites along simulated DNA minicircles and loops. Moreover, the placement of HU along loops with the "wild-type" spacing found in the Escherichia coli lactose (lac) and galactose (gal) operons precludes access to key recognition elements on DNA. The HU protein introduces a unique spatial pathway in the DNA upon closure. The many ways in which the protein induces nearly the same closed circular configuration point to the statistical advantage of its nonspecificity. The rotational settings imposed on DNA by the repressor proteins, by contrast, introduce sequential specificity in HU placement, with the nonspecific protein accumulating at particular loci on the constrained duplex. Thus, an architectural protein with no discernible DNA sequence-recognizing features becomes site-specific and potentially assumes a functional role upon loop formation. The locations of HU on the closed DNA reflect long-range mechanical correlations. The protein responds to DNA shape and deformability­the stiff, naturally straight double-helical structure­rather than to the unique features of the constituent base pairs. The structures of the simulated loops suggest that HU architecture, like nucleosomal architecture, which modulates the ability of regulatory proteins to recognize their binding sites in the context of chromatin, may influence repressor-operator interactions in the context of the bacterial nucleoid.

A One-Nearest-Neighbor Approach to Identify the Original Time of Infection Using Censored Baboon Sepsis Data.

OBJECTIVES: Sepsis therapies have proven to be elusive because of the difficulty of translating biologically sound and effective interventions in animal models to humans. A part of this problem originates from the fact that septic patients present at various times after the onset of sepsis, whereas the exact time of infection is controlled in animal models. We sought to determine whether data mining longitudinal physiologic data in a nonhuman primate model of Escherichia coli-induced sepsis could help inform the time of onset of infection. DESIGN: A nearest-neighbor approach was used to back cast the time of onset of infection in animal models of sepsis. Animal data were censored to simulate prospective monitoring at any moment along the septic infection. This was compared against an uncensored database to find the most similar animal in order to estimate the infection onset time. Leave-one-out cross-validation was used for validation. Biomarker selection was performed based on the criteria of estimation accuracy and/or ease of measurement. SETTING: Computational experimental on existing experimental data. SUBJECTS: Retrospective data from 33 septic baboons (Papio ursinus) subjected to Escherichia coli infusion. Validation was performed using 14 pigs that were subjected to surgically induced fecal peritonitis and 22 pigs that were subjected to lipopolysaccharide infusion. MEASUREMENTS AND MAIN RESULTS: Longitudinal physiologic and serum markers, time of death. The presence of uniquely changing biomarkers during septic infection enabled the estimation of infection onset time in the datasets. Various combinations of temporal biomarkers, such as WBC, oxygen content, mean arterial pressure, and heart rate, yielded estimation accuracies of up to 97.8%. The use of temporal vital signs and a single measurement of serum biomarkers yielded highly accurate estimates without the need for invasive measurements. Validation in the pig data revealed similar results despite the heterogeneity of multiple experimental cohorts. This suggests that the method may be effective if sufficiently similar subjects are present in the database. CONCLUSIONS: One nearest-neighbor analysis showed promise in accurately identifying the onset of infection given a database of known infection times and of sufficient breadth. We suggest that this approach is ready for evaluation within the clinical setting using human data.

The inflammatory response to influenza A virus (H1N1): An experimental and mathematical study.

Mortality from influenza infections continues as a global public health issue, with the host inflammatory response contributing to fatalities related to the primary infection. Based on Ordinary Differential Equation (ODE) formalism, a computational model was developed for the in-host response to influenza A virus, merging inflammatory, innate, adaptive and humoral responses to virus and linking severity of infection, the inflammatory response, and mortality. The model was calibrated using dense cytokine and cell data from adult BALB/c mice infected with the H1N1 influenza strain A/PR/8/34 in sublethal and lethal doses. Uncertainty in model parameters and disease mechanisms was quantified using Bayesian inference and ensemble model methodology that generates probabilistic predictions of survival, defined as viral clearance and recovery of the respiratory epithelium. The ensemble recovers the expected relationship between magnitude of viral exposure and the duration of survival, and suggests mechanisms primarily responsible for survival, which could guide the development of immuno-modulatory interventions as adjuncts to current anti-viral treatments. The model is employed to extrapolate from available data survival curves for the population and their dependence on initial viral aliquot. In addition, the model allows us to illustrate the positive effect of controlled inflammation on influenza survival.

A mathematical model of intrahost pneumococcal pneumonia infection dynamics in murine strains.

The seriousness of pneumococcal pneumonia in mouse models has been shown to depend both on bacterial serotype and murine strain. We here present a simple ordinary differential equation model of the intrahost immune response to bacterial pneumonia that is capable of capturing diverse experimentally determined responses of various murine strains. We discuss the main causes of such differences while accounting for the uncertainty in the estimation of model parameters. We model the bacterial population in both the lungs and blood, the cellular death caused by the infection, and the activation and immigration of phagocytes to the infected tissue. The ensemble model suggests that inter-strain differences in response to streptococcus pneumonia inoculation reside in the strength of nonspecific immune response and the rate of extrapulmonary phagocytosis.

A large-scale immuno-epidemiological simulation of influenza A epidemics.

BACKGROUND: Agent based models (ABM) are useful to explore population-level scenarios of disease spread and containment, but typically characterize infected individuals using simplified models of infection and symptoms dynamics. Adding more realistic models of individual infections and symptoms may help to create more realistic population level epidemic dynamics. METHODS: Using an equation-based, host-level mathematical model of influenza A virus infection, we develop a function that expresses the dependence of infectivity and symptoms of an infected individual on initial viral load, age, and viral strain phenotype. We incorporate this response function in a population-scale agent-based model of influenza A epidemic to create a hybrid multiscale modeling framework that reflects both population dynamics and individualized host response to infection. RESULTS: At the host level, we estimate parameter ranges using experimental data of H1N1 viral titers and symptoms measured in humans. By linearization of symptoms responses of the host-level model we obtain a map of the parameters of the model that characterizes clinical phenotypes of influenza infection and immune response variability over the population. At the population-level model, we analyze the effect of individualizing viral response in agent-based model by simulating epidemics across Allegheny County, Pennsylvania under both age-specific and age-independent severity assumptions. CONCLUSIONS: We present a framework for multi-scale simulations of influenza epidemics that enables the study of population-level effects of individual differences in infections and symptoms, with minimal additional computational cost compared to the existing population-level simulations.

Dynamical simulations of DNA supercoiling and compression.

In the present article, we summarize our recent studies of DNA dynamics using the generalized immersed boundary method. Our analysis of the effects of electrostatic repulsion on the dynamics of DNA supercoiling revealed that, after perturbation, a pre-twisted DNA collapses into a compact supercoiled configuration that is sensitive to the initial excess link and ionic strength of the solvent. A stochastic extension of the generalized immersed boundary method shows that DNA in solution subjected to a constant electric field is compressed into a configuration with smaller radius of gyration and smaller ellipticity ratio than those expected for such a molecule in a thermodynamic equilibrium.

Structural insights into the role of architectural proteins in DNA looping deduced from computer simulations.

Bacterial gene expression is regulated by DNA elements that often lie far apart along the genomic sequence, but come close together during genetic processing. The intervening residues form loops, which are organized by the binding of various proteins. For example, the Escherichia coli Lac repressor protein binds DNA operators, separated by 92 or 401 bp, and suppresses the formation of gene products involved in the metabolism of lactose. The system also includes several highly abundant architectural proteins, such as the histone-like (heat-unstable) HU protein, which severely deform the double helix upon binding. In order to gain a better understanding of how the naturally stiff DNA double helix forms the short loops detected in vivo, we have developed new computational methods to study the effects of various non-specific binding proteins on the three-dimensional configurational properties of DNA sequences. The present article surveys the approach that we use to generate ensembles of spatially constrained protein-decorated DNA structures (minicircles and Lac repressor-mediated loops) and presents some of the insights gained from the correspondence between computation and experiment about the potential contributions of architectural and regulatory proteins to DNA looping and gene expression.

Using a continuum model to predict closure time of gaps in intestinal epithelial cell layers.

A two-dimensional continuum model of collective cell migration is used to predict the closure of gaps in intestinal epithelial cell layers. The model assumes that cell migration is governed by lamellipodia formation, cell-cell adhesion, and cell-substrate adhesion. Model predictions of the gap edge position and complete gap closure time are compared with experimental measures from cell layer scratch assays (also called scratch wound assays). The goal of the study is to combine experimental observations with mathematical descriptions of cell motion to identify effects of gap shape and area on closure time and to propose a method that uses a simple measure (e.g., area) to predict overall gap closure time early in the closure process. Gap closure time is shown to increase linearly with increasing gap area; however, gaps of equal areas but different aspect ratios differ greatly in healing time. Previous methods that calculate overall healing time according to the absolute or percent change in gap area assume that the gap area changes at a constant rate and typically underestimate gap closure time. In this study, data from scratch assays suggest that the rate of change of area is proportional to the first power or square root power of area.

Unifying cardiovascular modelling with deep reinforcement learning for uncertainty aware control of sepsis treatment.

Sepsis is a potentially life-threatening inflammatory response to infection or severe tissue damage. It has a highly variable clinical course, requiring constant monitoring of the patient's state to guide the management of intravenous fluids and vasopressors, among other interventions. Despite decades of research, there's still debate among experts on optimal treatment. Here, we combine for the first time, distributional deep reinforcement learning with mechanistic physiological models to find personalized sepsis treatment strategies. Our method handles partial observability by leveraging known cardiovascular physiology, introducing a novel physiology-driven recurrent autoencoder, and quantifies the uncertainty of its own results. Moreover, we introduce a framework for uncertainty-aware decision support with humans in the loop. We show that our method learns physiologically explainable, robust policies, that are consistent with clinical knowledge. Further our method consistently identifies high-risk states that lead to death, which could potentially benefit from more frequent vasopressor administration, providing valuable guidance for future research.

Identification of Clinical Phenotypes in Septic Patients Presenting With Hypotension or Elevated Lactate.

Introduction: Targeted therapies for sepsis have failed to show benefit due to high variability among subjects. We sought to demonstrate different phenotypes of septic shock based solely on clinical features and show that these relate to outcome. Methods: A retrospective analysis was performed of a 1,023-subject cohort with early septic shock from the ProCESS trial. Twenty-three clinical variables at baseline were analyzed using hierarchical clustering, with consensus clustering used to identify and validate the ideal number of clusters in a derivation cohort of 642 subjects from 20 hospitals. Clusters were visualized using heatmaps over 0, 6, 24, and 72 h. Clinical outcomes were 14-day all-cause mortality and organ failure pattern. Cluster robustness was confirmed in a validation cohort of 381 subjects from 11 hospitals. Results: Five phenotypes were identified, each with unique organ failure patterns that persisted in time. By enrollment criteria, all patients had shock. The two high-risk phenotypes were characterized by distinct multi-organ failure patterns and cytokine signatures, with the highest mortality group characterized most notably by liver dysfunction and coagulopathy while the other group exhibited primarily respiratory failure, neurologic dysfunction, and renal dysfunction. The moderate risk phenotype was that of respiratory failure, while low-risk phenotypes did not have a high degree of additional organ failure. Conclusions: Sepsis phenotypes with distinct biochemical abnormalities may be identified by clinical characteristics alone and likely provide an opportunity for early clinical actionability and prognosis.

Continuum model of collective cell migration in wound healing and colony expansion.

Collective cell migration plays an important role during wound healing and embryo development. Although the exact mechanisms that coordinate such migration are still unknown, experimental studies of moving cell layers have shown that the primary interactions governing the motion of the layer are the force of lamellipodia, the adhesion of cells to the substrate, and the adhesion of cells to each other. Here, we derive a two-dimensional continuum mechanical model of cell-layer migration that is based on a novel assumption of elastic deformation of the layer and incorporates basic mechanical interactions of cells as well as cell proliferation and apoptosis. The evolution equations are solved numerically using a level set method. The model successfully reproduces data from two types of experiments: 1), the contraction of an enterocyte cell layer during wound healing; and 2), the expansion of a radially symmetric colony of MDCK cells, both in the edge migration velocity and in cell-layer density. In accord with experimental observations, and in contrast to reaction-diffusion models, this model predicts a partial wound closure if lamellipod formation is inhibited at the wound edge and gives implications of the effect of spatially restricted proliferation.

A simple mathematical model of signaling resulting from the binding of lipopolysaccharide with Toll-like receptor 4 demonstrates inherent preconditioning behavior.

The complex biology of Gram-negative bacterial lipopolysaccharide (LPS) is central to the acute inflammatory response in sepsis and related diseases. Repeated treatment with LPS can lead to desensitization or enhancement of subsequent responses both in vitro and in vivo (a phenomenon known as preconditioning). Previous computational studies have demonstrated a role for anti-inflammatory influences in this process (J. Day, J. Rubin, Y. Vodovotz, C.C. Chow, A. Reynolds, G. Clermont, A reduced mathematical model of the acute inflammatory response: II. Capturing scenarios of repeated endotoxin administration. J. Theor. Biol. 242 (2006) 237). Since LPS signals via Toll-like receptor 4 (TLR4), we created a simple mathematical model in order to address the role of this receptor in both the normal and preconditioned response to LPS. We created a non-linear system of ordinary differential equations, consisting of free LPS, free TLR4, bound complex LPS-TLR4, and an intracellular signaling cascade (lumped into a single variable). We simulate the effects of preconditioning by small and large repeated doses of LPS on the system, varying the timing of the doses as well as the rate of expression of TLR4. Our simulations suggest that a simplified model of LPS/TLR4 signaling can account for complex preconditioning phenomena without invoking a specific signaling inhibition mechanism, but rather based on the dynamics of the signaling response itself, as well as the timing and magnitude of the LPS stimuli.

Mesoscale modeling of multi-protein-DNA assemblies: the role of the catabolic activator protein in Lac-repressor-mediated looping.

DNA looping plays a key role in the regulation of the lac operon in Escherichia coli. The presence of a tightly bent loop (between sequentially distant sites of Lac repressor protein binding) purportedly hinders the binding of RNA polymerase and subsequent transcription of the genetic message. The unexpectedly favorable binding interaction of this protein-DNA assembly with the catabolic activator protein (CAP), a protein that also bends DNA and paradoxically facilitates the binding of RNA polymerase, stimulated extension of our base-pair level theory of DNA elasticity to the treatment of DNA loops formed in the presence of several proteins. Here we describe in detail a procedure to determine the structures and free energies of multi-protein-DNA assemblies and illustrate the predicted effects of CAP binding on the configurations of the wild-type 92-bp Lac repressor-mediated O3-O1 DNA loop. We show that the DNA loop adopts an antiparallel orientation in the most likely structure and that this loop accounts for the published experimental observation that, when CAP is bound to the loop, one of the arms of LacR binds to an alternative site that is displaced from the original site by 5 bp.

Slow-fast analysis of a multi-group asset flow model with implications for the dynamics of wealth.

The multi-group asset flow model is a nonlinear dynamical system originally developed as a tool for understanding the behavioral foundations of market phenomena such as flash crashes and price bubbles. In this paper we use a modification of this model to analyze the dynamics of a single-asset market in situations when the trading rates of investors (i.e., their desire to exchange stock for cash) are prescribed ahead of time and independent of the state of the market. Under the assumption of fast trading compared to the time-rate of change in the prescribed trading rates we decompose the dynamics of the system to fast and slow components. We use the model to derive a variety of observations regarding the dynamics of price and investors' wealth, and the dependence of these quantities on the prescribed trading rates. In particular, we show that strategies with constant trading rates, which represent the well-known constant-rebalanced portfolio (CRP) strategies, are optimal in the sense that they minimize investment risks. In contrast, we show that investors pursuing non-CRP strategies are at risk of loss of wealth, as a result of the slow system not being integrable in the sense that cyclic trading rates do not always result in periodic price variations.