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1.  Markov chain Monte Carlo based analysis of post-translationally modified VDAC gating kinetics 
The voltage-dependent anion channel (VDAC) is the main conduit for permeation of solutes (including nucleotides and metabolites) of up to 5 kDa across the mitochondrial outer membrane (MOM). Recent studies suggest that VDAC activity is regulated via post-translational modifications (PTMs). Yet the nature and effect of these modifications is not understood. Herein, single channel currents of wild-type, nitrosated, and phosphorylated VDAC are analyzed using a generalized continuous-time Markov chain Monte Carlo (MCMC) method. This developed method describes three distinct conducting states (open, half-open, and closed) of VDAC activity. Lipid bilayer experiments are also performed to record single VDAC activity under un-phosphorylated and phosphorylated conditions, and are analyzed using the developed stochastic search method. Experimental data show significant alteration in VDAC gating kinetics and conductance as a result of PTMs. The effect of PTMs on VDAC kinetics is captured in the parameters associated with the identified Markov model. Stationary distributions of the Markov model suggest that nitrosation of VDAC not only decreased its conductance but also significantly locked VDAC in a closed state. On the other hand, stationary distributions of the model associated with un-phosphorylated and phosphorylated VDAC suggest a reversal in channel conformation from relatively closed state to an open state. Model analyses of the nitrosated data suggest that faster reaction of nitric oxide with Cys-127 thiol group might be responsible for the biphasic effect of nitric oxide on basal VDAC conductance.
doi:10.3389/fphys.2014.00513
PMCID: PMC4292549  PMID: 25628567
mitochondria; VDAC; lipid bilayer; electrophysiology; nitrosation; phosphorylation; MCMC
2.  The Inferred Cardiogenic Gene Regulatory Network in the Mammalian Heart 
PLoS ONE  2014;9(6):e100842.
Cardiac development is a complex, multiscale process encompassing cell fate adoption, differentiation and morphogenesis. To elucidate pathways underlying this process, a recently developed algorithm to reverse engineer gene regulatory networks was applied to time-course microarray data obtained from the developing mouse heart. Approximately 200 genes of interest were input into the algorithm to generate putative network topologies that are capable of explaining the experimental data via model simulation. To cull specious network interactions, thousands of putative networks are merged and filtered to generate scale-free, hierarchical networks that are statistically significant and biologically relevant. The networks are validated with known gene interactions and used to predict regulatory pathways important for the developing mammalian heart. Area under the precision-recall curve and receiver operator characteristic curve are 9% and 58%, respectively. Of the top 10 ranked predicted interactions, 4 have already been validated. The algorithm is further tested using a network enriched with known interactions and another depleted of them. The inferred networks contained more interactions for the enriched network versus the depleted network. In all test cases, maximum performance of the algorithm was achieved when the purely data-driven method of network inference was combined with a data-independent, functional-based association method. Lastly, the network generated from the list of approximately 200 genes of interest was expanded using gene-profile uniqueness metrics to include approximately 900 additional known mouse genes and to form the most likely cardiogenic gene regulatory network. The resultant network supports known regulatory interactions and contains several novel cardiogenic regulatory interactions. The method outlined herein provides an informative approach to network inference and leads to clear testable hypotheses related to gene regulation.
doi:10.1371/journal.pone.0100842
PMCID: PMC4074065  PMID: 24971943
3.  Cerebrospinal fluid is drained primarily via the spinal canal and olfactory route in young and aged spontaneously hypertensive rats 
Background
Many aspects of CSF dynamics are poorly understood due to the difficulties involved in quantification and visualization. In particular, there is debate surrounding the route of CSF drainage. Our aim was to quantify CSF flow, volume, and drainage route dynamics in vivo in young and aged spontaneously hypertensive rats (SHR) using a novel contrast-enhanced computed tomography (CT) method.
Methods
ICP was recorded in young (2–5 months) and aged (16 months) SHR. Contrast was administered into the lateral ventricles bilaterally and sequential CT imaging was used to visualize the entire intracranial CSF system and CSF drainage routes. A customized contrast decay software module was used to quantify CSF flow at multiple locations.
Results
ICP was significantly higher in aged rats than in young rats (11.52 ± 2.36 mmHg, versus 7.04 ± 2.89 mmHg, p = 0.03). Contrast was observed throughout the entire intracranial CSF system and was seen to enter the spinal canal and cross the cribriform plate into the olfactory mucosa within 9.1 ± 6.1 and 22.2 ± 7.1 minutes, respectively. No contrast was observed adjacent to the sagittal sinus. There were no significant differences between young and aged rats in either contrast distribution times or CSF flow rates. Mean flow rates (combined young and aged) were 3.0 ± 1.5 μL/min at the cerebral aqueduct; 3.5 ± 1.4 μL/min at the 3rd ventricle; and 2.8 ± 0.9 μL/min at the 4th ventricle. Intracranial CSF volumes (and as percentage total brain volume) were 204 ± 97 μL (8.8 ± 4.3%) in the young and 275 ± 35 μL (10.8 ± 1.9%) in the aged animals (NS).
Conclusions
We have demonstrated a contrast-enhanced CT technique for measuring and visualising CSF dynamics in vivo. These results indicate substantial drainage of CSF via spinal and olfactory routes, but there was little evidence of drainage via sagittal sinus arachnoid granulations in either young or aged animals. The data suggests that spinal and olfactory routes are the primary routes of CSF drainage and that sagittal sinus arachnoid granulations play a minor role, even in aged rats with higher ICP.
doi:10.1186/2045-8118-11-12
PMCID: PMC4057524  PMID: 24932405
Computed tomography; Cerebrospinal fluid dynamics; Contrast; Spontaneously hypertensive rat; Intracranial pressure (ICP); Age; CSF; SHR
5.  Arterial Stiffening Provides Sufficient Explanation for Primary Hypertension 
PLoS Computational Biology  2014;10(5):e1003634.
Hypertension is one of the most common age-related chronic disorders, and by predisposing individuals for heart failure, stroke, and kidney disease, it is a major source of morbidity and mortality. Its etiology remains enigmatic despite intense research efforts over many decades. By use of empirically well-constrained computer models describing the coupled function of the baroreceptor reflex and mechanics of the circulatory system, we demonstrate quantitatively that arterial stiffening seems sufficient to explain age-related emergence of hypertension. Specifically, the empirically observed chronic changes in pulse pressure with age and the impaired capacity of hypertensive individuals to regulate short-term changes in blood pressure arise as emergent properties of the integrated system. The results are consistent with available experimental data from chemical and surgical manipulation of the cardio-vascular system. In contrast to widely held opinions, the results suggest that primary hypertension can be attributed to a mechanogenic etiology without challenging current conceptions of renal and sympathetic nervous system function.
Author Summary
Hypertension is highly age-related and affects more than 1 billion people worldwide. It is a major source of morbidity and mortality as it makes us more prone to experience heart failure, stroke, and kidney disease. Despite intense research efforts over several decades, there is still no consensus on what are the primary causes of this disorder. Here we present a computational physiology model which shows that the increase in arterial stiffness that follows with age is sufficient to account for an overwhelming amount of experimental and clinical data on hypertension. We demonstrate quantitatively that the stiffening causes the baroreceptors, the blood-pressure sensors located in the arterial wall, to misinform the highly complex machinery responsible for blood pressure regulation. This misinformation occurs because the baroreceptors are strain sensitive, not pressure sensitive, and with stiffening the aortic wall strain ceases to be a good proxy for aortic blood pressure. Contrary to wide-held conceptions, the blood pressure regulation may thus become compromised without any other detrimental physiological change of the regulatory machinery. Our results therefore suggest that arterial stiffness represents a major therapeutic target by which an otherwise intact physiological machinery may be exploited for blood pressure regulation.
doi:10.1371/journal.pcbi.1003634
PMCID: PMC4031054  PMID: 24853828
6.  Bridging the genotype–phenotype gap: what does it take? 
The Journal of Physiology  2013;591(Pt 8):2055-2066.
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.
doi:10.1113/jphysiol.2012.248864
PMCID: PMC3634519  PMID: 23401613
7.  Modeling Advection and Diffusion of Oxygen in Complex Vascular Networks 
Annals of biomedical engineering  2001;29(4):298-310.
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.
PMCID: PMC3915047  PMID: 11339327
Convection-diffusion Model; Hypoxia; Oxygen
8.  Analysis of cardiovascular dynamics in pulmonary hypertensive C57BL6/J mice 
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.
doi:10.3389/fphys.2013.00355
PMCID: PMC3858724  PMID: 24376421
pulmonary hypertension; cardiac mechanics; myofiber; stress; strain; re-modeling
9.  A computational analysis of the long-term regulation of arterial pressure 
F1000Research  2013;2:208.
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.
doi:10.12688/f1000research.2-208.v2
PMCID: PMC3886803  PMID: 24555102
10.  Theoretical models for coronary vascular biomechanics: Progress & challenges 
Progress in biophysics and molecular biology  2010;104(0):10.1016/j.pbiomolbio.2010.10.001.
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.
doi:10.1016/j.pbiomolbio.2010.10.001
PMCID: PMC3817728  PMID: 21040741
Vascular structure; Mechanics; Haemodynamics; Mass transport; Regulation; Adaptation; Mathematical and computational model; Multi-scale; Cellular mechanics; Integration
11.  Effect of P2X4 and P2X7 receptor antagonism on the pressure diuresis relationship in rats 
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.
doi:10.3389/fphys.2013.00305
PMCID: PMC3807716  PMID: 24187541
purinergic; ATP; kidney disease; renal injury; renal vascular resistance
12.  A computational analysis of the long-term regulation of arterial pressure 
F1000Research  2013;2:208.
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.
doi:10.12688/f1000research.2-208.v1
PMCID: PMC3886803  PMID: 24555102
13.  Inadvertent Occlusion of the Anterior Choroidal Artery Explains Infarct Variability in the Middle Cerebral Artery Thread Occlusion Stroke Model 
PLoS ONE  2013;8(9):e75779.
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.
doi:10.1371/journal.pone.0075779
PMCID: PMC3775728  PMID: 24069448
14.  Use of partition coefficients in flow-limited physiologically-based pharmacokinetic modeling 
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.
doi:10.1007/s10928-012-9252-6
PMCID: PMC3400708  PMID: 22639356
Physiologically-based pharmacokinetics; Flow-limited; Permeability-limited; Well-stirred tank; Compartmental modeling; Partition coefficient; Biophysical; Diazepam; Cotinine
15.  Identification of the kinetic mechanism of succinyl-CoA synthetase 
Bioscience Reports  2013;33(1):e00014.
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.
doi:10.1042/BSR20120069
PMCID: PMC3549575  PMID: 23088689
catalytic mechanism; dead-end binding; enzyme; succinyl-CoA; AIC, Akaike information criterion; SCS, succinyl-CoA synthetase; TCA, tricarboxylic acid
16.  Physiologically-based pharmacokinetic tissue compartment model selection in drug development and risk assessment 
Journal of Pharmaceutical Sciences  2011;101(1):424-435.
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.
doi:10.1002/jps.22768
PMCID: PMC3314064  PMID: 21968734
Pharmacokinetics; Physiological model; Well stirred model; Tissue partition; In silico modeling; Physicochemical; Singular perturbation; Asymptotic matching; Flow-limited; Compartmental modeling
17.  A Parallel Algorithm for Reverse Engineering of Biological Networks 
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.
doi:10.1039/c1ib00117e
PMCID: PMC3424073  PMID: 22080176
18.  Modeling to link regional myocardial work, metabolism and blood flows 
Annals of biomedical engineering  2012;40(11):2379-2398.
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.
doi:10.1007/s10439-012-0613-5
PMCID: PMC3471663  PMID: 22915334
Excitation-contraction coupling; coronary blood flow; cellular metabolism; phosphorylation potential; oxygenation; blood-tissue exchange processes
19.  Advection and Diffusion of Substances in Biological Tissues With Complex Vascular Networks 
Annals of biomedical engineering  2000;28(3):253-268.
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.
PMCID: PMC3483094  PMID: 10784090
Blood-tissue exchange kinetics; Capillary permeability; Cardiac capillary densities; Oxygen diffusion; Flow-limited transport; Systems impulse response; Transport function
20.  Development of appropriate equations for physiologically based pharmacokinetic modeling of permeability-limited and flow-limited transport 
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.
doi:10.1007/s10928-011-9200-x
PMCID: PMC3193163  PMID: 21584718
Physiologically based pharmacokinetics; Flow-limited; Permeability-limited; Well-stirred tank; Compartmental modeling; Singular perturbation
21.  Multiscale Modeling and Data Integration in the Virtual Physiological Rat Project 
Annals of Biomedical Engineering  2012;40(11):2365-2378.
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.
doi:10.1007/s10439-012-0611-7
PMCID: PMC3463790  PMID: 22805979
Semantic annotation; Model merging; Model repositories; Biomedical ontologies; Data dissemination; Model sharing; Mechanistic physiological models; Virtual Physiological Rat
22.  Fractal15O-Labeled Water Washout From the Heart 
Circulation Research  1995;77(6):1212-1221.
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.
PMCID: PMC3362128  PMID: 7586234
flow-limited blood-tissue exchange; power law kinetics; positron emission; oxygen-15; capillary permeability; statistical self-similar processes
23.  Mechanisms of pressure-diuresis and pressure-natriuresis in Dahl salt-resistant and Dahl salt-sensitive rats 
BMC Physiology  2012;12:6.
Background
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.
Results
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.
Conclusions
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.
doi:10.1186/1472-6793-12-6
PMCID: PMC3536597  PMID: 22583378
24.  Apparent Diffusivity and Taylor Dispersion of Water and Solutes in Capillary Beds 
Bulletin of Mathematical Biology  2009;71(6):1366-1377.
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.
doi:10.1007/s11538-009-9405-y
PMCID: PMC3305791  PMID: 19234745
25.  Simulation of Cellular Biochemical System Kinetics 
The goal of realistically and reliably simulating the biochemical processes underlying cellular function is achievable through a systematic approach that makes use of the broadest possible amount of in vitro and in vivo data, and is consistent with all applicable physical chemical theory. Progress will be facilitated by establishing: (1.) a concrete self-consistent theoretical foundation for systems simulation; (2.) extensive and accurate databases of thermodynamic properties of biochemical reactions; (3.) parameterized and validated models of enzyme and transporter catalytic mechanisms that are consistent with physical chemical theoretical foundation; and (4.) software tools for integrating all of these concepts, data, and models into a cohesive representation of cellular biochemical systems. Ongoing initiatives are laying the groundwork for the broad-based community cooperation that will be necessary to pursue these elements of a strategic infrastructure for systems simulation on a large scale.
doi:10.1002/wsbm.116
PMCID: PMC3293220  PMID: 21171044

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