The genotype–phenotype map (GP map) concept applies to any time point in the ontogeny of a living system. It is the outcome of very complex dynamics that include environmental effects, and bridging the genotype–phenotype gap is synonymous with understanding these dynamics. The context for this understanding is physiology, and the disciplinary goals of physiology do indeed demand the physiological community to seek this understanding. We claim that this task is beyond reach without use of mathematical models that bind together genetic and phenotypic data in a causally cohesive way. We provide illustrations of such causally cohesive genotype–phenotype models where the phenotypes span from gene expression profiles to development of whole organs. Bridging the genotype–phenotype gap also demands that large-scale biological (‘omics’) data and associated bioinformatics resources be more effectively integrated with computational physiology than is currently the case. A third major element is the need for developing a phenomics technology way beyond current state of the art, and we advocate the establishment of a Human Phenome Programme solidly grounded on biophysically based mathematical descriptions of human physiology.
A realistic geometric model for the three-dimensional capillary network geometry is used as a framework for studying the transport and consumption of oxygen in cardiac tissue. The nontree-like capillary network conforms to the available morphometric statistics and is supplied by a single arterial source and drains into a pair of venular sinks. We explore steady-state oxygen transport and consumption in the tissue using a mathematical model which accounts for advection in the vascular network, nonlinear binding of dissolved oxygen to hemoglobin and myoglobin, passive diffusion of freely dissolved and protein-bound oxygen, and Michaelis–Menten consumption in the parenchymal tissue. The advection velocity field is found by solving the hemodynamic problem for flow throughout the network. The resulting system is described by a set of coupled nonlinear elliptic equations, which are solved using a finite-difference numerical approximation. We find that coupled advection and diffusion in the three-dimensional system enhance the dispersion of oxygen in the tissue compared to the predictions of simplified axially distributed models, and that no “lethal corner,” or oxygen-deprived region occurs for physiologically reasonable values for flow and consumption. Concentrations of 0.5–1.0 mM myoglobin facilitate the transport of oxygen and thereby protect the tissue from hypoxia at levels near its p50, that is, when local oxygen consumption rates are close to those of delivery by flow and myoglobin-facilitated diffusion, a fairly narrow range.
Convection-diffusion Model; Hypoxia; Oxygen
A computer model was used to analyze data on cardiac and vascular mechanics from C57BL6/J mice exposed to 0 (n = 4), 14 (n = 6), 21 (n = 8) and 28 (n = 7) days of chronic hypoxia and treatment with the VEGF receptor inhibitor SUGEN (HySu) to induce pulmonary hypertension. Data on right ventricular pressure and volume, and systemic arterial pressure obtained before, during, and after inferior vena cava occlusion were analyzed using a mathematical model of realistic ventricular mechanics coupled with a simple model of the pulmonary and systemic vascular systems. The model invokes a total of 26 adjustable parameters, which were estimated based on least-squares fitting of the data. Of the 26 adjustable parameters, 14 were set to globally constant values for the entire data set. It was necessary to adjust the remaining 12 parameters to match data from all experimental groups. Of these 12 individually adjusted parameters, three parameters representing pulmonary vascular resistance, pulmonary arterial elastance, and pulmonary arterial narrowing were found to significantly change in HySu-induced remodeling. Model analysis shows a monotonic change in these parameters as disease progressed, with approximately 130% increase in pulmonary resistance, 70% decrease in unstressed pulmonary arterial volume, and 110% increase in pulmonary arterial elastance in the 28-day group compared to the control group. These changes are consistent with prior experimental measurements. Furthermore, the 28-day data could be explained only after increasing the passive elastance of the right free wall compared to the value used for the other data sets, which is likely a consequence of the increased RV collagen accumulation found experimentally. These findings may indicate a compensatory remodeling followed by pathological remodeling of the right ventricle in HySu-induced pulmonary hypertension.
pulmonary hypertension; cardiac mechanics; myofiber; stress; strain; re-modeling
The asserted dominant role of the kidneys in the chronic regulation of blood pressure and in the etiology of hypertension has been debated since the 1970s. At the center of the theory is the observation that the acute relationships between arterial pressure and urine production—the acute pressure-diuresis and pressure-natriuresis curves—physiologically adapt to perturbations in pressure and/or changes in the rate of salt and volume intake. These adaptations, modulated by various interacting neurohumoral mechanisms, result in chronic relationships between water and salt excretion and pressure that are much steeper than the acute relationships. While the view that renal function is the dominant controller of arterial pressure has been supported by computer models of the cardiovascular system known as the “Guyton-Coleman model”, no unambiguous description of a computer model capturing chronic adaptation of acute renal function in blood pressure control has been presented. Here, such a model is developed with the goals of: 1. representing the relevant mechanisms in an identifiable mathematical model; 2. identifying model parameters using appropriate data; 3. validating model predictions in comparison to data; and 4. probing hypotheses regarding the long-term control of arterial pressure and the etiology of primary hypertension. The developed model reveals: long-term control of arterial blood pressure is primarily through the baroreflex arc and the renin-angiotensin system; and arterial stiffening provides a sufficient explanation for the etiology of primary hypertension associated with ageing. Furthermore, the model provides the first consistent explanation of the physiological response to chronic stimulation of the baroreflex.
A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.
Vascular structure; Mechanics; Haemodynamics; Mass transport; Regulation; Adaptation; Mathematical and computational model; Multi-scale; Cellular mechanics; Integration
Reduced glomerular filtration, hypertension and renal microvascular injury are hallmarks of chronic kidney disease, which has a global prevalence of ~10%. We have shown previously that the Fischer (F344) rat has lower GFR than the Lewis rat, and is more susceptible to renal injury induced by hypertension. In the early stages this injury is limited to the pre-glomerular vasculature. We hypothesized that poor renal hemodynamic function and vulnerability to vascular injury are causally linked and genetically determined. In the present study, normotensive F344 rats had a blunted pressure diuresis relationship, compared with Lewis rats. A kidney microarray was then interrogated using the Endeavour enrichment tool to rank candidate genes for impaired blood pressure control. Two novel candidate genes, P2rx7 and P2rx4, were identified, having a 7− and 3− fold increased expression in F344 rats. Immunohistochemistry localized P2X4 and P2X7 receptor expression to the endothelium of the pre-glomerular vasculature. Expression of both receptors was also found in the renal tubule; however there was no difference in expression profile between strains. Brilliant Blue G (BBG), a relatively selective P2X7 antagonist suitable for use in vivo, was administered to both rat strains. In Lewis rats, BBG had no effect on blood pressure, but increased renal vascular resistance, consistent with inhibition of some basal vasodilatory tone. In F344 rats BBG caused a significant reduction in blood pressure and a decrease in renal vascular resistance, suggesting that P2X7 receptor activation may enhance vasoconstrictor tone in this rat strain. BBG also reduced the pressure diuresis threshold in F344 rats, but did not alter its slope. These preliminary findings suggest a physiological and potential pathophysiological role for P2X7 in controlling renal and/or systemic vascular function, which could in turn affect susceptibility to hypertension-related kidney damage.
purinergic; ATP; kidney disease; renal injury; renal vascular resistance
The asserted dominant role of the kidneys in the chronic regulation of blood pressure and in the etiology of hypertension has been debated since the 1970s. At the center of the theory is the observation that the acute relationships between arterial pressure and urine production—the acute pressure-diuresis and pressure-natriuresis curves—physiologically adapt to perturbations in pressure and/or changes in the rate of salt and volume intake. These adaptations, modulated by various interacting neurohumoral mechanisms, result in chronic relationships between water and salt excretion and pressure that are much steeper than the acute relationships. While the view that renal function is the dominant controller of arterial pressure has been supported by computer models of the cardiovascular system known as the “Guyton-Coleman model”, no unambiguous description of a computer model capturing chronic adaptation of acute renal function in blood pressure control has been presented. Here, such a model is developed with the goals of: 1. capturing the relevant mechanisms in an identifiable mathematical model; 2. identifying model parameters using appropriate data; 3. validating model predictions in comparison to data; and 4. probing hypotheses regarding the long-term control of arterial pressure and the etiology of primary hypertension. The developed model reveals: long-term control of arterial blood pressure is primarily through the baroreflex arc and the renin-angiotensin system; and arterial stiffening provides a sufficient explanation for the etiology of primary hypertension associated with ageing. Furthermore, the model provides the first consistent explanation of the physiological response to chronic stimulation of the baroreflex.
Intraluminal occlusion of the middle cerebral artery (MCAo) in rodents is perhaps the most widely used model of stroke, however variability of infarct volume and the ramifications of this on sample sizes remains a problem, particularly for preclinical testing of potential therapeutics. Our data and that of others, has shown a dichotomous distribution of infarct volumes for which there had previously been no clear explanation. When studying perfusion computed tomography cerebral blood volume (CBV) maps obtained during intraluminal MCAo in rats, we observed inadvertent occlusion of the anterior choroidal artery (AChAo) in a subset of animals. We hypothesized that the combined occlusion of the MCA and AChA may be a predictor of larger infarct volume following stroke. Thus, we aimed to determine the correlation between AChAo and final infarct volume in rats with either temporary or permanent MCA occlusion (1 h, 2 h, or permanent MCAo). Outbred Wistar rats (n = 28) were imaged prior to and immediately following temporary or permanent middle cerebral artery occlusion. Presence of AChAo on CBV maps was shown to be a strong independent predictor of 24 h infarct volume (β = 0.732, p <0.001). This provides an explanation for the previously observed dichotomous distribution of infarct volumes. Interestingly, cortical infarct volumes were also larger in rats with AChAo, although the artery does not supply cortex. This suggests an important role for perfusion of the MCA territory beyond the proximal occlusion through AChA-MCA anastomotic collateral vessels in animals with a patent AChAo. Identification of combined MCAo and AChAo will allow other investigators to tailor their stroke model to reduce variability in infarct volumes, improve statistical power and reduce sample sizes in preclinical stroke research.
Permeability-limited two-subcompartment and flow-limited, well-stirred tank tissue compartment models are routinely used in physiologically-based pharmacokinetic modeling. Here, the permeability-limited two-subcompartment model is used to derive a general flow-limited case of a two-subcompartment model with the well-stirred tank being a specific case where tissue fractional blood volume approaches zero. The general flow-limited two-subcompartment model provides a clear distinction between two partition coefficients typically used in PBPK: a biophysical partition coefficient and a well-stirred partition coefficient. Case studies using diazepam and cotinine demonstrate that, when the well-stirred tank is used with a priori predicted biophysical partition coefficients, simulations overestimate or underestimate total organ drug concentration relative to flow-limited two-subcompartment model behavior in tissues with higher fractional blood volumes. However, whole-body simulations show predicted drug concentrations in plasma and lower fractional blood volume tissues are relatively unaffected. These findings point to the importance of accurately determining tissue fractional blood volume for flow-limited PBPK modeling. Simulations using biophysical and well-stirred partition coefficients optimized with flow-limited two-subcompartment and well-stirred models, respectively, lead to nearly identical fits to tissue drug distribution data. Therefore, results of whole-body PBPK modeling with diazepam and cotinine indicate both flow-limited models are appropriate PBPK tissue models as long as the correct partition coefficient is used: the biophysical partition coefficient is for use with two-subcompartment models and the well-stirred partition coefficient is for use with the well-stirred tank model.
Physiologically-based pharmacokinetics; Flow-limited; Permeability-limited; Well-stirred tank; Compartmental modeling; Partition coefficient; Biophysical; Diazepam; Cotinine
Standard Gibbs energies of reactions are increasingly being used in metabolic modeling for applying thermodynamic constraints on reaction rates, metabolite concentrations and kinetic parameters. The increasing scope and diversity of metabolic models has led scientists to look for genome-scale solutions that can estimate the standard Gibbs energy of all the reactions in metabolism. Group contribution methods greatly increase coverage, albeit at the price of decreased precision. We present here a way to combine the estimations of group contribution with the more accurate reactant contributions by decomposing each reaction into two parts and applying one of the methods on each of them. This method gives priority to the reactant contributions over group contributions while guaranteeing that all estimations will be consistent, i.e. will not violate the first law of thermodynamics. We show that there is a significant increase in the accuracy of our estimations compared to standard group contribution. Specifically, our cross-validation results show an 80% reduction in the median absolute residual for reactions that can be derived by reactant contributions only. We provide the full framework and source code for deriving estimates of standard reaction Gibbs energy, as well as confidence intervals, and believe this will facilitate the wide use of thermodynamic data for a better understanding of metabolism.
The metabolism of living organisms is a complex system with a large number of parameters and interactions. Nevertheless, it is governed by a strict set of rules that make it somewhat predictable and amenable to modeling. The laws of thermodynamics play a pivotal role by determining reaction feasibility and by governing the kinetics of enzymes. Here we introduce estimations for the standard Gibbs energy of reactions, with the best combination of accuracy and coverage to date. The estimations are derived using a new method which we denote component contribution. This method integrates multiple sources of information into a consistent framework that obeys the laws of thermodynamics, and provides a significant improvement in accuracy compared to previous genome-wide estimations of standard Gibbs energies. We apply and test our method on reconstructions of E. coli and human metabolism and, in addition, do our best to facilitate the use of these estimations in future models by providing open-source software that performs the integration in a streamlined process.
The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper.
Diagnostic markers are derived from steady state measurements, but are used to predict the cellular response to therapy. To develop new and better diagnostics, we would like to systematically characterize the relationship between steady state and the response to a given therapeutic. Mathematical models have powerfully complemented empirical studies in this regard, but it remains challenging to employ these models to characterize the effects of steady state. To do so requires a mathematical expression for the steady state, for which no universal method has been developed. Here, we present a method for deriving a mathematical expression for the steady state of a common class of models, those that obey the Law of Mass Action. We show that our method is easy to use and scales well to large models. We then use our method to characterize the relationship between steady state and the sensitivity to the anti-cancer drug candidate, dulanermin. We find that sensitivity to the drug is strongly affected by the concentration of the signaling molecule, procaspase 8, and its inhibitor, Bar. Our work thus demonstrates the utility of analytical studies of the steady state and its relationship to drug sensitivity.
The kinetic mechanism of SCS [succinyl-CoA (coenzyme A) synthetase], which participates in the TCA (tricarboxylic acid) cycle, ketone body metabolism and haem biosynthesis, has not been fully characterized. Namely, a representative catalytic mechanism and associated kinetic parameters that can explain data on the enzyme-catalysed reaction kinetics have not been established. To determine an accurate model, a set of putative mechanisms of SCS, proposed by previous researchers, were tested against experimental data (from previous publication) on SCS derived from porcine myocardium. Based on comparisons between model simulation and the experimental data, an ordered ter–ter mechanism with dead-end product inhibition of succinate against succinyl-CoA is determined to be the best candidate mechanism. A thermodynamically constrained set of parameter values is identified for this candidate mechanism.
catalytic mechanism; dead-end binding; enzyme; succinyl-CoA; AIC, Akaike information criterion; SCS, succinyl-CoA synthetase; TCA, tricarboxylic acid
The well-stirred tank (WST) has been the predominant flow-limited tissue compartment model in physiologically-based pharmacokinetic (PBPK) modeling. Recently, we developed a two-region asymptotically reduced (TAR) PBPK tissue compartment model through an asymptotic approximation to a two-region vascular-extravascular system to incorporate more biophysical detail than the WST model. To determine the relevance of the novel flow-limited approach (F-TAR), 75 structurally diverse drugs are evaluated herein using a priori predicted tissue:plasma partition coefficients along with hybrid and whole-body PBPK of eight rat tissues to determine the impact of model selection on simulation and optimization. Simulations show the F-TAR model significantly improves the ability to predict drug exposure, with hybrid and whole-body WST model error approaching 50% for tissues with larger vascular volumes. When optimization is used to fit F-TAR and WST models to pseudo data, WST-optimized drug partition coefficients more appropriately represent curve-fitting parameters rather than biophysically meaningful partition coefficients. Median F-TAR-optimized error ranged from -0.4 to 0.3%, while WST-optimized median error ranged from -22.2 to 1.8%. These studies demonstrate the use of F-TAR represents a more accurate, biophysically realistic PBPK tissue model for predicting tissue exposure to drug and should be considered for use in drug development and regulatory review.
Pharmacokinetics; Physiological model; Well stirred model; Tissue partition; In silico modeling; Physicochemical; Singular perturbation; Asymptotic matching; Flow-limited; Compartmental modeling
Dynamic biological systems, such as gene regulatory networks (GRNs) and protein signaling networks, are often represented as systems of ordinary differential equations. Such equations can be utilized in reverse engineering these biological networks, specifically since identifying these networks is challenging due to the cost of the necessary experiments growing with at least the square of the size of the system. Moreover, the number of possible models, proportional to the number of directed graphs connecting nodes representing the variables in the system, suffers from combinatorial explosion as the size of the system grows. Therefore, exhaustive searches for systems of nontrivial complexity are not feasible. Here we describe a practical and scalable algorithm for determining candidate network interactions based on decomposing an N-dimensional system into N one-dimensional problems. The algorithm was tested on in silico networks based on known biological GRNs. The computational complexity of the network identification is shown to increase as N2 while a parallel implementation achieves essentially linear speedup with the increasing number of processing cores. For each in silico network tested, the algorithm successfully predicts a candidate network that reproduces the network dynamics. This approach dramatically reduces the computational demand required for reverse engineering GRNs and produces a wealth of exploitable information in the process. Moreover, the candidate network topologies returned by the algorithm can be used to design future experiments aimed at gathering informative data capable of further resolving the true network topology.
Given the mono-functional, highly coordinated processes of cardiac excitation and contraction, the observations that regional myocardial blood flows, rMBF, are broadly heterogeneous has provoked much attention, but a clear explanation has not emerged. In isolated and in vivo heart studies the total coronary flow is found to be proportional to the rate-pressure product (systolic mean blood pressure times heart rate), a measure of external cardiac work. The same relationship might be expected on a local basis: more work requires more flow. The validity of this expectation has never been demonstrated experimentally. In this article we review the concepts linking cellular excitation and contractile work to cellular energetics and ATP demand, substrate utilization, oxygen demand, vasoregulation, and local blood flow. Mathematical models of these processes are now rather well developed. We propose that the construction of an integrated model encompassing the biophysics, biochemistry and physiology of cardiomyocyte contraction, then combined with a detailed three-dimensional structuring of the fiber bundle and sheet arrangements of the heart as a whole will frame an hypothesis that can be quantitatively evaluated to settle the prime issue: Does local work drive local flow in a predictable fashion that explains the heterogeneity? While in one sense one can feel content that work drives flow is irrefutable, there are no cardiac contractile models that demonstrate the required heterogeneity in local strain-stress-work; quite the contrary, cardiac contraction models have tended toward trying to show that work should be uniform. The object of this review is to argue that uniformity of work does not occur, and is impossible in any case, and that further experimentation and analysis are necessary to test the hypothesis.
Excitation-contraction coupling; coronary blood flow; cellular metabolism; phosphorylation potential; oxygenation; blood-tissue exchange processes
For highly diffusive solutes the kinetics of blood–tissue exchange is only poorly represented by a model consisting of sets of independent parallel capillary–tissue units. We constructed a more realistic multicapillary network model conforming statistically to morphometric data. Flows through the tortuous paths in the network were calculated based on constant resistance per unit length throughout the network and the resulting advective intracapillary velocity field was used as a framework for describing the extravascular diffusion of a substance for which there is no barrier or permeability limitation. Simulated impulse responses from the system, analogous to tracer water outflow dilution curves, showed flow-limited behavior over a range of flows from about 2 to 5 ml min−1 g−1, as is observed for water in the heart in vivo. The present model serves as a reference standard against which to evaluate computationally simpler, less physically realistic models. The simulated outflow curves from the network model, like experimental water curves, were matched to outflow curves from the commonly used axially distributed models only by setting the capillary wall permeability–surface area (PS) to a value so artifactually low that it is incompatible with the experimental observations that transport is flow limited. However, simple axially distributed models with appropriately high PSs will fit water outflow dilution curves if axial diffusion coefficients are set at high enough values to account for enhanced dispersion due to the complex geometry of the capillary network. Without incorporating this enhanced dispersion, when applied to experimental curves over a range of flows, the simpler models give a false inference that there is recruitment of capillary surface area with increasing flow. Thus distributed models must account for diffusional as well as permeation processes to provide physiologically appropriate parameter estimates.
Blood-tissue exchange kinetics; Capillary permeability; Cardiac capillary densities; Oxygen diffusion; Flow-limited transport; Systems impulse response; Transport function
Mitochondria form a dynamic tubular reticulum within eukaryotic cells. Currently, quantitative understanding of its morphological characteristics is largely absent, despite major progress in deciphering the molecular fission and fusion machineries shaping its structure. Here we address the principles of formation and the large-scale organization of the cell-wide network of mitochondria. On the basis of experimentally determined structural features we establish the tip-to-tip and tip-to-side fission and fusion events as dominant reactions in the motility of this organelle. Subsequently, we introduce a graph-based model of the chondriome able to encompass its inherent variability in a single framework. Using both mean-field deterministic and explicit stochastic mathematical methods we establish a relationship between the chondriome structural network characteristics and underlying kinetic rate parameters. The computational analysis indicates that mitochondrial networks exhibit a percolation threshold. Intrinsic morphological instability of the mitochondrial reticulum resulting from its vicinity to the percolation transition is proposed as a novel mechanism that can be utilized by cells for optimizing their functional competence via dynamic remodeling of the chondriome. The detailed size distribution of the network components predicted by the dynamic graph representation introduces a relationship between chondriome characteristics and cell function. It forms a basis for understanding the architecture of mitochondria as a cell-wide but inhomogeneous organelle. Analysis of the reticulum adaptive configuration offers a direct clarification for its impact on numerous physiological processes strongly dependent on mitochondrial dynamics and organization, such as efficiency of cellular metabolism, tissue differentiation and aging.
Mitochondria control energy production, initiation of cell death and several other critical cellular processes. Most often, they form a constantly reshaping tubular reticulum spread over the cytosol. Despite extensive knowledge of mitochondrial physiology, accurate description of their large-scale architecture is missing, partly due to substantial variability of reticulum geometries found in different cell types, and a capability for fast radical changes. We address this shortcoming with a mathematical model representing the organelle as a cell-wide dynamical network subjected to opposing actions of fission and internal fusion – processes known experimentally but not yet accurately specified. This opens a way for the quantitative characterization of the large-scale organization by showing how particular types of the internal dynamics can shape the reticulum into the whole multitude of configurations observed in biological studies. Further analysis reveals that for a specific value of tip-to-side fission/fusion rates the network should undergo a radical reorganization. Because of the high morphological sensitivity to minute changes in fusion or fission rates close to the critical point, cells can quickly adapt the mitochondrial operation and structure to their actual needs at a low expenditure of energy.
Although the implementation of a flow-limited, well-stirred tank (WST) single-compartment tissue model in pharmacokinetics and toxicokinetics is widespread, its use is not always justified biophysically or physiologically. The WST model introduces a loss of biophysical detail, specifically the vascular space, which is present in the standard permeability-limited two-subcompartment (PLT) tissue model. To address this loss of detail when evaluating the in vivo kinetics of drugs, toxins, nutrients, and endogenous metabolites, a novel set of physiologically based pharmacokinetic tissue compartment equations is developed through application of an asymptotic approximation to a two-region vascular–extravascular system to arrive at a permeability-limited two-region asymptotically reduced (P-TAR) model and a flow-limited (F-TAR) model. Development of the TAR modeling approach illustrates the importance of relative timescales in PBPK tissue compartment model selection and the conditions under which improved biophysical realism is advantageous. In the permeability-limited regime, the TAR model formulations enable drug or toxicant concentration to be modeled in the vascular and extravascular spaces equivalent to the PLT tissue model while invoking only one state variable to represent the vascular and extravascular spaces. In the flow-limited regime, the F-TAR model is more biophysically realistic than the WST model because it maintains the anatomical distinction between the vascular and extravascular spaces, and hence offers greater pharmacological and physiological insight than the WST model, without introducing additional computational complexity.
Physiologically based pharmacokinetics; Flow-limited; Permeability-limited; Well-stirred tank; Compartmental modeling; Singular perturbation
It has become increasingly evident that the descriptions of many complex diseases are only possible by taking into account multiple influences at different physiological scales. To do this with computational models often requires the integration of several models that have overlapping scales (genes to molecules, molecules to cells, cells to tissues). The Virtual Physiological Rat (VPR) Project, a National Institute of General Medical Sciences (NIGMS) funded National Center of Systems Biology, is tasked with mechanistically describing several complex diseases and is therefore identifying methods to facilitate the process of model integration across physiological scales. In addition, the VPR has a considerable experimental component and the resultant data must be integrated into these composite multiscale models and made available to the research community. A perspective of the current state of the art in model integration and sharing along with archiving of experimental data will be presented here in the context of multiscale physiological models. It was found that current ontological, model and data repository resources and integrative software tools are sufficient to create composite models from separate existing models and the example composite model developed here exhibits emergent behavior not predicted by the separate models.
Semantic annotation; Model merging; Model repositories; Biomedical ontologies; Data dissemination; Model sharing; Mechanistic physiological models; Virtual Physiological Rat
Mathematical models that integrate multi-scale physiological data can offer insight into physiological and pathophysiological function, and may eventually assist in individualized predictive medicine. We present a methodology for performing systematic analyses of multi-parameter interactions in such complex, multi-scale models. Human physiology models are often based on or inspired by Arthur Guyton's whole-body circulatory regulation model. Despite the significance of this model, it has not been the subject of a systematic and comprehensive sensitivity study. Therefore, we use this model as a case study for our methodology. Our analysis of the Guyton model reveals how the multitude of model parameters combine to affect the model dynamics, and how interesting combinations of parameters may be identified. It also includes a “virtual population” from which “virtual individuals” can be chosen, on the basis of exhibiting conditions similar to those of a real-world patient. This lays the groundwork for using the Guyton model for in silico exploration of pathophysiological states and treatment strategies. The results presented here illustrate several potential uses for the entire dataset of sensitivity results and the “virtual individuals” that we have generated, which are included in the supplementary material. More generally, the presented methodology is applicable to modern, more complex multi-scale physiological models.
As our understanding of the human body at all scales increases, the construction of a “Virtual Physiological Human” is becoming more feasible and will be an important step towards individualized diagnosis and treatment. As computational models increase in complexity to reflect this growth in understanding, analysis of these models becomes ever more complex. We present a methodology for systematically analysing the interactions between parameters and outputs of such complicated models, using the Guyton model of circulatory regulation as a case study. This model remains a landmark achievement that contributed to the development of our current understanding of blood pressure control, and we present the first comprehensive sensitivity analysis of this model. Effects of varying each parameter are explored over randomized simulations; our analysis demonstrates how to use these results to infer relationships between model parameters and the predicted physiological behaviour. Understanding these relationships is of the utmost importance for developing an optimal treatment strategy for individual patients. These results provide new insight into the multi-level interactions in the cardiovascular-renal system and will be useful to researchers wishing to use the model in pathophysiological or pharmacological settings. This methodology is applicable to current and future physiological models of arbitrary complexity.
The integration of various types of genomic data into predictive models of biological networks is one of the main challenges currently faced by computational biology. Constraint-based models in particular play a key role in the attempt to obtain a quantitative understanding of cellular metabolism at genome scale. In essence, their goal is to frame the metabolic capabilities of an organism based on minimal assumptions that describe the steady states of the underlying reaction network via suitable stoichiometric constraints, specifically mass balance and energy balance (i.e. thermodynamic feasibility). The implementation of these requirements to generate viable configurations of reaction fluxes and/or to test given flux profiles for thermodynamic feasibility can however prove to be computationally intensive. We propose here a fast and scalable stoichiometry-based method to explore the Gibbs energy landscape of a biochemical network at steady state. The method is applied to the problem of reconstructing the Gibbs energy landscape underlying metabolic activity in the human red blood cell, and to that of identifying and removing thermodynamically infeasible reaction cycles in the Escherichia coli metabolic network (iAF1260). In the former case, we produce consistent predictions for chemical potentials (or log-concentrations) of intracellular metabolites; in the latter, we identify a restricted set of loops (23 in total) in the periplasmic and cytoplasmic core as the origin of thermodynamic infeasibility in a large sample () of flux configurations generated randomly and compatibly with the prior information available on reaction reversibility.
The operation of biological systems is constrained under all circumstances by the laws of physics. Thermodynamics, in particular, dictates preferential directions in which biochemical reactions should flow at stationarity. When applied to cellular reaction systems (like metabolic networks), it favors the emergence of some (thermodynamically feasible) ways to organize the flow of matter while prohibiting others. The development of detailed predictive models for the biochemical activity of a cell relies on the possibility to integrate the laws of thermodynamics in genome-scale reconstructions of cellular metabolic networks. In this work we have devised an efficient relaxation algorithm to implement thermodynamic constraints in genome-scale models. Besides allowing to check for thermodynamic feasibility of reaction flow configurations, it is also capable of providing information on other relevant physico-chemical quantities. We have applied it to two cellular metabolic networks of different complexity, namely that of human red blood cells and that of the bacterium Escherichia coli. In the former case, we have obtained predictions for the intracellular chemical state (in terms of metabolite concentrations and reaction free energies) consistent with empirical knowledge; in the latter, we have effectively corrected thermodynamically infeasible flux configurations.
To characterize the washout of water from the heart, we used a flow-limited (not diffusion- or permeability-limited) marker for blood-tissue exchange, namely, tracer-labeled water. Experiments were performed by injecting 15O-labeled water into the inflow to isolated blood-perfused rabbit hearts and by recording the tracer content in the heart and in the outflow simultaneously for up to 5 minutes. The data exhibit a particular combination of power law forms: (1) The downslopes of the residue and outflow curves were both power law functions, with the residue diminishing as t−α and the outflow as t−α−1, where α is interpreted to be the dimensionless exponent of a fractal power law relation characterizing the selfsimilarity inherent in each curve. (2) The fractional escape rate, given by the outflow curve divided by the residue curve, diminished almost exactly as t−l. In 18 sets of curves, α averaged 2.21 ± 0.27. These results lead to an improved method for extrapolating the downslopes of indicator dilution curves to estimate their areas and therefore the blood flows. The evidence also points strongly to the conclusions that myocardial water washout is a fractal process and that stirred tank models are inappropriate for the heart.
flow-limited blood-tissue exchange; power law kinetics; positron emission; oxygen-15; capillary permeability; statistical self-similar processes
Data on blood flow regulation, renal filtration, and urine output in salt-sensitive Dahl S rats fed on high-salt (hypertensive) and low-salt (prehypertensive) diets and salt-resistant Dahl R rats fed on high-salt diets were analyzed using a mathematical model of renal blood flow regulation, glomerular filtration, and solute transport in a nephron.
The mechanism of pressure-diuresis and pressure-natriuresis that emerges from simulation of the integrated systems is that relatively small increases in glomerular filtration that follow from increases in renal arterial pressure cause relatively large increases in urine and sodium output. Furthermore, analysis reveals the minimal differences between the experimental cases necessary to explain the observed data. It is determined that differences in renal afferent and efferent arterial resistances are able to explain all of the qualitative differences in observed flows, filtration rates, and glomerular pressure as well as the differences in the pressure-natriuresis and pressure-diuresis relationships in the three groups. The model is able to satisfactorily explain data from all three groups without varying parameters associated with glomerular filtration or solute transport in the nephron component of the model.
Thus the differences between the experimental groups are explained solely in terms of difference in blood flow regulation. This finding is consistent with the hypothesis that, if a shift in the pressure-natriuresis relationship is the primary cause of elevated arterial pressure in the Dahl S rat, then alternation in how renal afferent and efferent arterial resistances are regulated represents the primary cause of chronic hypertension in the Dahl S rat.
A decade ago, a team of biochemists including two of us, modeled yeast glycolysis and showed that one of the most studied biochemical pathways could not be quite understood in terms of the kinetic properties of the constituent enzymes as measured in cell extract. Moreover, when the same model was later applied to different experimental steady-state conditions, it often exhibited unrestrained metabolite accumulation.
Here we resolve this issue by showing that the results of such ab initio modeling are improved substantially by (i) including appropriate allosteric regulation and (ii) measuring the enzyme kinetic parameters under conditions that resemble the intracellular environment. The following modifications proved crucial: (i) implementation of allosteric regulation of hexokinase and pyruvate kinase, (ii) implementation of Vmax values measured under conditions that resembled the yeast cytosol, and (iii) redetermination of the kinetic parameters of glyceraldehyde-3-phosphate dehydrogenase under physiological conditions.
Model predictions and experiments were compared under five different conditions of yeast growth and starvation. When either the original model was used (which lacked important allosteric regulation), or the enzyme parameters were measured under conditions that were, as usual, optimal for high enzyme activity, fructose 1,6-bisphosphate and some other glycolytic intermediates tended to accumulate to unrealistically high concentrations. Combining all adjustments yielded an accurate correspondence between model and experiments for all five steady-state and dynamic conditions. This enhances our understanding of in vivo metabolism in terms of in vitro biochemistry.
Baker's yeast is widely applied in modern biotechnology, for instance for production of heterologous protein or biofuel. For such applications a thorough understanding of the central energy metabolism of the bug is crucial. Nevertheless, even for this well-known organism, attempts to build models ab initio, based on independently measured characteristics of the catalysts (the enzymes), seldom gives reliable results. A key problem in this field is that enzyme characteristics are often studied under non-physiological conditions that do not resemble the environment inside the cell. In this study we measured the enzyme characteristics under physiological conditions and assembled the results into a computational model of yeast energy metabolism. We show that this simple trick greatly improves the predictive value of the computational model. This allowed us to predict correctly how yeast cells adapt to nitrogen starvation, an industrially relevant situation, in which remodeling of the proteome strongly affects cellular energy metabolism.
A physical theory explaining the anisotropic dispersion of water and solutes in biological tissues is introduced based on the phenomena of Taylor dispersion, in which highly diffusive solutes cycle between flowing and stagnant regions in the tissue, enhancing dispersion in the direction of microvascular flow. An effective diffusion equation is derived, for which the coefficient of dispersion in the axial direction (direction of capillary orientation) depends on the molecular diffusion coefficient, tissue perfusion, and vessel density. This analysis provides a homogenization that represents three-dimensional transport in capillary beds as an effectively one-dimensional phenomenon. The derived dispersion equation may be used to simulate the transport of solutes in tissues, such as in pharmacokinetic modeling. In addition, the analysis provides a physically based hypothesis for explaining dispersion anisotropy observed in diffusion-weighted imaging (DWI) and diffusion-tensor magnetic resonance imaging (DTMRI) and suggests a means of obtaining quantitative functional information on capillary vessel density from measurements of dispersion coefficients. It is shown that a failure to account for flow-mediated dispersion in vascular tissues may lead to misinterpretations of imaging data and significant overestimates of directional bias in molecular diffusivity in biological tissues. Measurement of the ratio of axial to transverse diffusivity may be combined with an independent measurement of perfusion to provide an estimate of capillary vessel density in the tissue.