Modelling spatio-temporal interactions within the cell

Smeared Multiscale Finite Element Models for Mass Transport and Electrophysiology Coupled to Muscle Mechanics

Mass transport represents the most fundamental process in living organisms. It includes delivery of nutrients, oxygen, drugs, and other substances from the vascular system to tissue and transport of waste and other products from cells back to vascular and lymphatic network and organs. Furthermore, movement is achieved by mechanical forces generated by muscles in coordination with the nervous system. The signals coming from the brain, which have the character of electrical waves, produce activation within muscle cells. Therefore, from a physics perspective, there exist a number of physical fields within the body, such as velocities of transport, pressures, concentrations of substances, and electrical potential, which is directly coupled to biochemical processes of transforming the chemical into mechanical energy and further internal forces for motion. The overall problems of mass transport and electrophysiology coupled to mechanics can be investigated theoretically by developing appropriate computational models. Due to the enormous complexity of the biological system, it would be almost impossible to establish a detailed computational model for the physical fields related to mass transport, electrophysiology, and coupled fields. To make computational models feasible for applications, we here summarize a concept of smeared physical fields, with coupling among them, and muscle mechanics, which includes dependence on the electrical potential. Accuracy of the smeared computational models, also with coupling to muscle mechanics, is illustrated with simple example, while their applicability is demonstrated on a liver model with tumors present. The last example shows that the introduced methodology is applicable to large biological systems.

A predictive computational model reveals that GIV/girdin serves as a tunable valve for EGFR-stimulated cyclic AMP signals

Cellular levels of the versatile second messenger cyclic (c)AMP are regulated by the antagonistic actions of the canonical G protein → adenylyl cyclase pathway that is initiated by G-protein–coupled receptors (GPCRs) and attenuated by phosphodiesterases (PDEs). Dysregulated cAMP signaling drives many diseases; for example, its low levels facilitate numerous sinister properties of cancer cells. Recently, an alternative paradigm for cAMP signaling has emerged in which growth factor–receptor tyrosine kinases (RTKs; e.g., EGFR) access and modulate G proteins via a cytosolic guanine-nucleotide exchange modulator (GEM), GIV/girdin; dysregulation of this pathway is frequently encountered in cancers. In this study, we present a network-based compartmental model for the paradigm of GEM-facilitated cross-talk between RTKs and G proteins and how that impacts cellular cAMP. Our model predicts that cross-talk between GIV, Gα_s, and Gα_i proteins dampens ligand-stimulated cAMP dynamics. This prediction was experimentally verified by measuring cAMP levels in cells under different conditions. We further predict that the direct proportionality of cAMP concentration as a function of receptor number and the inverse proportionality of cAMP concentration as a function of PDE concentration are both altered by GIV levels. Taking these results together, our model reveals that GIV acts as a tunable control valve that regulates cAMP flux after growth factor stimulation. For a given stimulus, when GIV levels are high, cAMP levels are low, and vice versa. In doing so, GIV modulates cAMP via mechanisms distinct from the two most often targeted classes of cAMP modulators, GPCRs and PDEs.

Multiscale smeared finite element model for mass transport in biological tissue: From blood vessels to cells and cellular organelles

One of the basic and vital processes in living organisms is mass exchange, which occurs on several levels: it goes from blood vessels to cells and organelles within cells. On that path, molecules, as oxygen, metabolic products, drugs, etc. Traverse different macro and micro environments - blood, extracellular/intracellular space, and interior of organelles; and also biological barriers such as walls of blood vessels and membranes of cells and organelles. Many aspects of this mass transport remain unknown, particularly the biophysical mechanisms governing drug delivery. The main research approach relies on laboratory and clinical investigations. In parallel, considerable efforts have been directed to develop computational tools for additional insight into the intricate process of mass exchange and transport. Along these lines, we have recently formulated a composite smeared finite element (CSFE) which is composed of the smeared continuum pressure and concentration fields of the capillary and lymphatic system, and of these fields within tissue. The element offers an elegant and simple procedure which opens up new lines of inquiry and can be applied to large systems such as organs and tumors models. Here, we extend this concept to a multiscale scheme which concurrently couples domains that span from large blood vessels, capillaries and lymph, to cell cytosol and further to organelles of nanometer size. These spatial physical domains are coupled by the appropriate connectivity elements representing biological barriers. The composite finite element has "degrees of freedom" which include pressures and concentrations of all compartments of the vessels-tissue assemblage. The overall model uses the standard, measurable material properties of the continuum biological environments and biological barriers. It can be considered as a framework into which we can incorporate various additional effects (such as electrical or biochemical) for transport through membranes or within cells. This concept and the developed FE software within our package PAK offers a computational tool that can be applied to whole-organ systems, while also including specific domains such as tumors. The solved examples demonstrate the accuracy of this model and its applicability to large biological systems.

Stability analysis in spatial modeling of cell signaling

Advances in high-resolution microscopy and other techniques have emphasized the spatio-temporal nature of information transfer through signal transduction pathways. The compartmentalization of signaling molecules and the existence of microdomains are now widely acknowledged as key features in biochemical signaling. To complement experimental observations of spatio-temporal dynamics, mathematical modeling has emerged as a powerful tool. Using modeling, one can not only recapitulate experimentally observed dynamics of signaling molecules, but also gain an understanding of the underlying mechanisms in order to generate experimentally testable predictions. Reaction-diffusion systems are commonly used to this end; however, the analysis of coupled nonlinear systems of partial differential equations, generated by considering large reaction networks is often challenging. Here, we aim to provide an introductory tutorial for the application of reaction-diffusion models to the spatio-temporal dynamics of signaling pathways. In particular, we outline the steps for stability analysis of such models, with a focus on biochemical signal transduction. WIREs Syst Biol Med 2018, 10:e1395. doi: 10.1002/wsbm.1395 This article is categorized under: Biological Mechanisms > Cell Signaling Analytical and Computational Methods > Dynamical Methods Models of Systems Properties and Processes > Mechanistic Models.

A Probabilistic Boolean Network Approach for the Analysis of Cancer-Specific Signalling: A Case Study of Deregulated PDGF Signalling in GIST

BACKGROUND

Signal transduction networks are increasingly studied with mathematical modelling approaches while each of them is suited for a particular problem. For the contextualisation and analysis of signalling networks with steady-state protein data, we identified probabilistic Boolean network (PBN) as a promising framework which could capture quantitative changes of molecular changes at steady-state with a minimal parameterisation.

RESULTS AND CONCLUSION

In our case study, we successfully applied the PBN approach to model and analyse the deregulated Platelet-Derived Growth Factor (PDGF) signalling pathway in Gastrointestinal Stromal Tumour (GIST). We experimentally determined a rich and accurate dataset of steady-state profiles of selected downstream kinases of PDGF-receptor-alpha mutants in combination with inhibitor treatments. Applying the tool optPBN, we fitted a literature-derived candidate network model to the training dataset consisting of single perturbation conditions. Model analysis suggested several important crosstalk interactions. The validity of these predictions was further investigated experimentally pointing to relevant ongoing crosstalk from PI3K to MAPK signalling in tumour cells. The refined model was evaluated with a validation dataset comprising multiple perturbation conditions. The model thereby showed excellent performance allowing to quantitatively predict the combinatorial responses from the individual treatment results in this cancer setting. The established optPBN pipeline is also widely applicable to gain a better understanding of other signalling networks at steady-state in a context-specific fashion.

A G-protein subunit translocation embedded network motif underlies GPCR regulation of calcium oscillations

G-protein βγ subunits translocate reversibly from the plasma membrane to internal membranes on receptor activation. Translocation rates differ depending on the γ subunit type. There is limited understanding of the role of the differential rates of Gβγ translocation in modulating signaling dynamics in a cell. Bifurcation analysis of the calcium oscillatory network structure predicts that the translocation rate of a signaling protein can regulate the damping of system oscillation. Here, we examined whether the Gβγ translocation rate regulates calcium oscillations induced by G-protein-coupled receptor activation. Oscillations in HeLa cells expressing γ subunit types with different translocation rates were imaged and quantitated. The results show that differential Gβγ translocation rates can underlie the diversity in damping characteristics of calcium oscillations among cells. Mathematical modeling shows that a translocation embedded motif regulates damping of G-protein-mediated calcium oscillations consistent with experimental data. The current study indicates that such a motif may act as a tuning mechanism to design oscillations with varying damping patterns by using intracellular translocation of a signaling component.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

Modeling and Simulation of Intracellular Drug Transport and Disposition Pathways with Virtual Cell

The development of computational approaches for modeling the spatiotemporal dynamics of intracellular, small molecule drug concentrations has become an increasingly important area of pharmaceutical research. For systems pharmacology, the system dynamics of subcellular transport can be coupled to downstream pharmacological effects on biochemical pathways that impact cell structure and function. Here, we demonstrate how a widely used systems biology modeling package - Virtual Cell - can also be used to model the intracellular, passive transport pathways of small druglike molecules. Using differential equations to represent passive drug transport across cellular membranes, spatiotemporal changes in the intracellular distribution and concentrations of exogenous chemical agents in specific subcellular organelles were simulated for weakly acidic, neutral, and basic molecules, as a function of the molecules' lipophilicity and ionization potentials. In addition, we simulated the transport properties of small molecule chemical agents in the presence of a homogenous extracellular concentration or a transcellular concentration gradient. We also simulated the effects of cell type-dependent variations in the intracellular microenvironments on the distribution and accumulation of small molecule chemical agents in different organelles over time, under influx and efflux conditions. Lastly, we simulated the transcellular transport of small molecule chemical agents, in the presence of different apical and basolateral microenvironments. By incorporating existing models of drug permeation and subcellular distribution, our results indicate that Virtual Cell can provide a user-friendly, open, online computational modeling platform for systems pharmacology and biopharmaceutics research, making mathematical models and simulation results accessible to a broad community of users, without requiring advanced computer programming knowledge.

A spatial physiological model for p53 intracellular dynamics

In this paper we design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of ATM by DNA damage. The p53 protein is known to have a central role in the response of the cell to cytotoxic or radiotoxic insults resulting in DNA damage. A reasonable requirement for a model describing intracellular signalling pathways is taking into account the basic feature of eukaryotic cells: the distinction between nucleus and cytoplasm. Our aim is to show, on a simple reaction network describing p53 dynamics, how this basic distinction provides a framework which is able to yield expected oscillatory dynamics without introducing either positive feedbacks or delays in the reactions. Furthermore we prove that oscillations appear only if some spatial constraints are respected, e.g. if the diffusion coefficients correspond to known biological values. Finally we analyse how the spatial features of a cell influence the dynamic response of the p53 network to DNA damage, pointing out that the protein oscillatory dynamics is indeed a response that is robust towards changes with respect to cellular environments. Even if we change the cell shape or its volume or better its ribosomal distribution, we observe that DNA damage yields sustained oscillations of p53.

Influence of the nuclear membrane, active transport, and cell shape on the Hes1 and p53-Mdm2 pathways: insights from spatio-temporal modelling

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53-Mdm2, both of which are implicated in cancer. In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15-31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53-Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

The natural and engineered 3D microenvironment as a regulatory cue during stem cell fate determination

The concept of using stem cells as self-renewing sources of healthy cells in regenerative medicine has existed for decades, but most applications have yet to achieve clinical success. A main reason for the lack of successful stem cell therapies is the difficulty in fully recreating the maintenance and control of the native stem cell niche. Improving the performance of transplanted stem cells therefore requires a better understanding of the cellular mechanisms guiding stem cell behavior in both native and engineered three-dimensional (3D) microenvironments. Most techniques, however, for uncovering mechanisms controlling cell behavior in vitro have been developed using 2D cell cultures and are of limited use in 3D environments such as engineered tissue constructs. Deciphering the mechanisms controlling stem cell fate in native and engineered 3D environments, therefore, requires rigorous quantitative techniques that permit mechanistic, hypothesis-driven studies of cell-microenvironment interactions. Here, we review the current understanding of 2D and 3D stem cell control mechanisms and propose an approach to uncovering the mechanisms that govern stem cell behavior in 3D.

Extraction of elementary rate constants from global network analysis of E. coli central metabolism

Background

As computational performance steadily increases, so does interest in extending one-particle-per-molecule models to larger physiological problems. Such models however require elementary rate constants to calculate time-dependent rate coefficients under physiological conditions. Unfortunately, even when in vivo kinetic data is available, it is often in the form of aggregated rate laws (ARL) that do not specify the required elementary rate constants corresponding to mass-action rate laws (MRL). There is therefore a need to develop a method which is capable of automatically transforming ARL kinetic information into more detailed MRL rate constants.

Results

By incorporating proteomic data related to enzyme abundance into an MRL modelling framework, here we present an efficient method operating at a global network level for extracting elementary rate constants from experiment-based aggregated rate law (ARL) models. The method combines two techniques that can be used to overcome the difficult properties in parameterization. The first, a hybrid MRL/ARL modelling technique, is used to divide the parameter estimation problem into sub-problems, so that the parameters of the mass action rate laws for each enzyme are estimated in separate steps. This reduces the number of parameters that have to be optimized simultaneously. The second, a hybrid algebraic-numerical simulation and optimization approach, is used to render some rate constants identifiable, as well as to greatly narrow the bounds of the other rate constants that remain unidentifiable. This is done by incorporating equality constraints derived from the King-Altman and Cleland method into the simulated annealing algorithm. We apply these two techniques to estimate the rate constants of a model of E. coli glycolytic pathways. The simulation and statistical results show that our innovative method performs well in dealing with the issues of high computation cost, stiffness, local minima and uncertainty inherent with large-scale non-convex nonlinear MRL models.

Conclusion

In short, this new hybrid method can ensure the proper solution of a challenging parameter estimation problem of nonlinear dynamic MRL systems, while keeping the computational effort reasonable. Moreover, the work provides us with some optimism that physiological models at the particle scale can be rooted on a firm foundation of parameters generated in the macroscopic regime on an experimental basis. Thus, the proposed method should have applications to multi-scale modelling of the real biological systems allowing for enzyme intermediates, stochastic and spatial effects inside a cell.