Histochemical demonstration of glycogen phosphorylase (E.C. 2.4.11) through the use of semipermeable membranes (author's transl)]

A method is described for the histochemical demonstration of phosphorylase in unfixed cryostat sections of rat liver using semipermeable membranes and a gel medium. In comparison to the conventional methods this procedure has the following advantages: 1. Staining of sections through a semipermeable membrane prevents diffusion of cellular glycogen and guarantees optimal localisation of phosphorylase activity. 2. Since diffusion is effectively prevented by the membrane the total activity of this highly soluble enzyme can be demonstrated. 3. Tissue and cell structures are well preserved.     

Mesoscopic effects in an agent-based bargaining model in regular lattices

The effect of spatial structure has been proved very relevant in repeated games. In this work we propose an agent based model where a fixed finite population of tagged agents play iteratively the Nash demand game in a regular lattice. The model extends the multiagent bargaining model by Axtell, Epstein and Young modifying the assumption of global interaction. Each agent is endowed with a memory and plays the best reply against the opponent's most frequent demand. We focus our analysis on the transient dynamics of the system, studying by computer simulation the set of states in which the system spends a considerable fraction of the time. The results show that all the possible persistent regimes in the global interaction model can also be observed in this spatial version. We also find that the mesoscopic properties of the interaction networks that the spatial distribution induces in the model have a significant impact on the diffusion of strategies, and can lead to new persistent regimes different from those found in previous research. In particular, community structure in the intratype interaction networks may cause that communities reach different persistent regimes as a consequence of the hindering diffusion effect of fluctuating agents at their borders.     

Mathematical Conditions for Induced Cell Differentiation and Trans-differentiation in Adult Cells

We present a theoretical framework for the analysis of the effect of a fully differentiated cell population on a neighboring stem cell population in Multi-Cellular Organisms (MCOs). Such an organism is constituted by a set of different cell populations, each set of which converges to a different cycle from all possible options, of the same Boolean network. Cells communicate via a subset of the nodes called signals. We show that generic dynamic properties of cycles and nodes in random Boolean networks can induce cell differentiation. Specifically we propose algorithms, conditions and methods to examine if a set of signaling nodes enabling these conversions can be found. Surprisingly we find that robust conversions can be obtained even with a very small number of signals. The proposed conversions can occur in multiple spatial organizations and can be used as a model for regeneration in MCOs, where an islet of cells of one type (representing stem cells) is surrounded by cells of another type (representing differentiated cells). The cells at the outer layer of the islet function like progenitor cells (i.e. dividing asymmetrically and differentiating). To the best of our knowledge, this is the first work showing a tissue-like regeneration in MCO simulations based on random Boolean networks. We show that the probability to obtain a conversion decreases with the log of the node number in the network, showing that the model is relevant for large networks as well. We have further checked that the conversions are not trivial, i.e. conversions do not occur due to irregular structures of the Boolean network, and the converting cycle undergoes a respectable change in its behavior. Finally we show that the model can also be applied to a realistic genetic regulatory network, showing that the basic mathematical insight from regular networks holds in more complex experiment-based networks.     

Random-walk models of cell dispersal included in mechanobiological simulations of tissue differentiation

Computational models have shown that biophysical stimuli can be correlated with observed patterns of tissue differentiation, and simulations have been performed that predict the time course of tissue differentiation in, for example, long bone fracture healing. Some simulations have used a diffusion model to simulate the migration and proliferation of cells with the differentiating tissue. However, despite the convenience of the diffusion model, diffusion is not the mechanism of cell dispersal: cells disperse by crawling or proliferation, or are transported in a moving fluid. In this paper, a random-walk model (i.e., a stochastic model), with and without a preferred direction, is studied as an approach to simulate cell proliferation/migration in differentiating tissues and it is compared with the diffusion model. A simulation of tissue differentiation of gap tissue in a two-dimensional model of a bone/implant interface was performed to demonstrate the differences between diffusion vs. random walk with a preferred direction. Results of diffusion and random-walk models are similar with respect to the change in the stiffness of the gap tissue but rather different results are obtained regarding tissue patterning in the differentiating tissues; the diffusion approach predicted continuous patterns of tissue differentiation whereas the random-walk model showed a more discontinuous pattern-histological results are not available that can unequivocally establish which is most similar to experimental observation. Comparing isotropic to anisotropic random walk (preferred direction of proliferation and cell migration), a more rapid reduction of the relative displacement between implant and bone is predicted. In conclusion, we have shown how random-walk models of cell dispersal and proliferation can be implemented, and shown where differences between them exist. Further study of the random-walk model is warranted, given the importance of cell seeding and cell dispersal/proliferation in many mechanobiological problems.     

A Mass Conserved Reaction–Diffusion System Captures Properties of Cell Polarity

Cell polarity is a general cellular process that can be seen in various cell types such as migrating neutrophils and Dictyostelium cells. The Rho small GTP(guanosine 5′-tri phosphate)ases have been shown to regulate cell polarity; however, its mechanism of emergence has yet to be clarified. We first developed a reaction–diffusion model of the Rho GTPases, which exhibits switch-like reversible response to a gradient of extracellular signals, exclusive accumulation of Cdc42 and Rac, or RhoA at the maximal or minimal intensity of the signal, respectively, and tracking of changes of a signal gradient by the polarized peak. The previous cell polarity models proposed by Subramanian and Narang show similar behaviors to our Rho GTPase model, despite the difference in molecular networks. This led us to compare these models, and we found that these models commonly share instability and a mass conservation of components. Based on these common properties, we developed conceptual models of a mass conserved reaction–diffusion system with diffusion–driven instability. These conceptual models retained similar behaviors of cell polarity in the Rho GTPase model. Using these models, we numerically and analytically found that multiple polarized peaks are unstable, resulting in a single stable peak (uniqueness of axis), and that sensitivity toward changes of a signal gradient is specifically restricted at the polarized peak (localized sensitivity). Although molecular networks may differ from one cell type to another, the behaviors of cell polarity in migrating cells seem similar, suggesting that there should be a fundamental principle. Thus, we propose that a mass conserved reaction–diffusion system with diffusion-driven instability is one of such principles of cell polarity.     

Statistics of weighted brain networks reveal hierarchical organization and Gaussian degree distribution

Whole brain weighted connectivity networks were extracted from high resolution diffusion MRI data of 14 healthy volunteers. A statistically robust technique was proposed for the removal of questionable connections. Unlike most previous studies our methods are completely adapted for networks with arbitrary weights. Conventional statistics of these weighted networks were computed and found to be comparable to existing reports. After a robust fitting procedure using multiple parametric distributions it was found that the weighted node degree of our networks is best described by the normal distribution, in contrast to previous reports which have proposed heavy tailed distributions. We show that post-processing of the connectivity weights, such as thresholding, can influence the weighted degree asymptotics. The clustering coefficients were found to be distributed either as gamma or power-law distribution, depending on the formula used. We proposed a new hierarchical graph clustering approach, which revealed that the brain network is divided into a regular base-2 hierarchical tree. Connections within and across this hierarchy were found to be uncommonly ordered. The combined weight of our results supports a hierarchically ordered view of the brain, whose connections have heavy tails, but whose weighted node degrees are comparable.     

Facilitated diffusion in chromatin lattices: mechanistic diversity and regulatory potential

The interaction between a protein and a specific DNA site is the molecular basis for vital processes in all organisms. Location of the DNA target site by the protein commonly involves facilitated diffusion. Mechanisms of facilitated diffusion vary among proteins; they include one- and two-dimensional sliding along DNA, direct transfer between uncorrelated sites, as well as combinations of these mechanisms. Facilitated diffusion has almost exclusively been studied in vitro. This review discusses facilitated diffusion in the context of the living cell and proposes a theoretical model for facilitated diffusion in chromatin lattices. Chromatin structure differentially affects proteins in different modes of diffusion. The interplay of facilitated diffusion and chromatin structure can determine the rate of protein association with the target site, the frequency of association-dissociation events at the target site, and, under particular conditions, the occupancy of the target site. Facilitated diffusion is required in vivo for efficient DNA repair and bacteriophage restriction and has potential roles in fine-tuning gene regulatory networks and kinetically compartmentalizing the eukaryotic nucleus.     

Accounting for diffusion in agent based models of reaction-diffusion systems with application to cytoskeletal diffusion

Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface.     

Particle Transport through Hydrogels Is Charge Asymmetric

Transport processes within biological polymer networks, including mucus and the extracellular matrix, play an important role in the human body, where they serve as a filter for the exchange of molecules and nanoparticles. Such polymer networks are complex and heterogeneous hydrogel environments that regulate diffusive processes through finely tuned particle-network interactions. In this work, we present experimental and theoretical studies to examine the role of electrostatics on the basic mechanisms governing the diffusion of charged probe molecules inside model polymer networks. Translational diffusion coefficients are determined by fluorescence correlation spectroscopy measurements for probe molecules in uncharged as well as cationic and anionic polymer solutions. We show that particle transport in the charged hydrogels is highly asymmetric, with diffusion slowed down much more by electrostatic attraction than by repulsion, and that the filtering capability of the gel is sensitive to the solution ionic strength. Brownian dynamics simulations of a simple model are used to examine key parameters, including interaction strength and interaction range within the model networks. Simulations, which are in quantitative agreement with our experiments, reveal the charge asymmetry to be due to the sticking of particles at the vertices of the oppositely charged polymer networks.     

Life as physics and chemistry: A system view of biology

Cellular life can be viewed as one of many physical natural systems that extract free energy from their environments in the most efficient way, according to fundamental physical laws, and grow until limited by inherent physical constraints. Thus, it can be inferred that it is the efficiency of this process that natural selection acts upon. The consequent emphasis on metabolism, rather than replication, points to a metabolism-first origin of life with the adoption of DNA template replication as a second stage development. This order of events implies a cellular regulatory system that pre-dates the involvement of DNA and might, therefore, be based on the information acquired as peptides fold into proteins, rather than on genetic regulatory networks. Such an epigenetic cell regulatory model, the independent attractor model, has already been proposed to explain the phenomenon of radiation induced genomic instability. Here it is extended to provide an epigenetic basis for the morphological and functional diversity that evolution has yielded, based on natural selection of the most efficient free energy transduction. Empirical evidence which challenges the current genetic basis of cell and molecular biology and which supports the above proposal is discussed.     

The quantitative basis for the redistribution of immobile bacterial lipoproteins to division septa

The spatial localisation of proteins is critical for most cellular function. In bacteria, this is typically achieved through capture by established landmark proteins. However, this requires that the protein is diffusive on the appropriate timescale. It is therefore unknown how the localisation of effectively immobile proteins is achieved. Here, we investigate the localisation to the division site of the slowly diffusing lipoprotein Pal, which anchors the outer membrane to the cell wall of Gram-negative bacteria. While the proton motive force-linked TolQRAB system is known to be required for this repositioning, the underlying mechanism is unresolved, especially given the very low mobility of Pal. We present a quantitative, mathematical model for Pal relocalisation in which dissociation of TolB-Pal complexes, powered by the proton motive force across the inner membrane, leads to the net transport of Pal along the outer membrane and its deposition at the division septum. We fit the model to experimental measurements of protein mobility and successfully test its predictions experimentally against mutant phenotypes. Our model not only explains a key aspect of cell division in Gram-negative bacteria, but also presents a physical mechanism for the transport of low-mobility proteins that may be applicable to multi-membrane organelles, such as mitochondria and chloroplasts.     

Mathematical modelling and computational study of two-dimensional and three-dimensional dynamics of receptor–ligand interactions in signalling response mechanisms

Cell signalling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signalling pathways. But the binding process still presents many enigmas and, by analogy with surface catalytic reactions, two different mechanisms can be conceived: the first mechanism is related to the Eley–Rideal (ER) mechanism, i.e. the bulk-dissolved ligand interacts directly by pure three-dimensional (3D) diffusion with the specific surface receptor; the second mechanism is similar to the Langmuir–Hinshelwood (LH) process, i.e. 3D diffusion of the ligand to the cell surface followed by reversible ligand adsorption and subsequent two-dimensional (2D) surface diffusion to the receptor. A situation where both mechanisms simultaneously contribute to the signalling process could also occur. The aim of this paper is to perform a computational study of the behavior of the signalling response when these different mechanisms for ligand-receptor interactions are integrated into a model for signal transduction and ligand transport. To this end, partial differential equations have been used to develop spatio-temporal models that show trafficking dynamics of ligands, cell surface components, and intracellular signalling molecules through the different domains of the system. The mathematical modeling developed for these mechanisms has been applied to the study of two situations frequently found in cell systems: (a) dependence of the signal response on cell density; and (b) enhancement of the signalling response in a synaptic environment.     

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.     

Protein diffusion in mammalian cell cytoplasm

We introduce a new method for mesoscopic modeling of protein diffusion in an entire cell. This method is based on the construction of a three-dimensional digital model cell from confocal microscopy data. The model cell is segmented into the cytoplasm, nucleus, plasma membrane, and nuclear envelope, in which environment protein motion is modeled by fully numerical mesoscopic methods. Finer cellular structures that cannot be resolved with the imaging technique, which significantly affect protein motion, are accounted for in this method by assigning an effective, position-dependent porosity to the cell. This porosity can also be determined by confocal microscopy using the equilibrium distribution of a non-binding fluorescent protein. Distinction can now be made within this method between diffusion in the liquid phase of the cell (cytosol/nucleosol) and the cytoplasm/nucleoplasm. Here we applied the method to analyze fluorescence recovery after photobleach (FRAP) experiments in which the diffusion coefficient of a freely-diffusing model protein was determined for two different cell lines, and to explain the clear difference typically observed between conventional FRAP results and those of fluorescence correlation spectroscopy (FCS). A large difference was found in the FRAP experiments between diffusion in the cytoplasm/nucleoplasm and in the cytosol/nucleosol, for all of which the diffusion coefficients were determined. The cytosol results were found to be in very good agreement with those by FCS.     

Uptake and localisation of small-molecule fluorescent probes in living cells: a critical appraisal of QSAR models and a case study concerning probes for DNA and RNA

Small-molecule fluorochromes are used in biology and medicine to generate informative microscopic and macroscopic images, permitting identification of cell structures, measurement of physiological/physicochemical properties, assessment of biological functions and assay of chemical components. Modes of uptake and precise intracellular localisation of a probe are typically significant factors in its successful application. These processes and localisations can be predicted using quantitative structure activity relations (QSAR) models, which correlate aspects of the physicochemical properties of the probes (expressed numerically) with the uptake/localisation. Pay-offs of such modelling include better understanding and trouble-shooting of current and novel probes, and easier design of future probes (“guided synthesis”). Uptake models discussed consider adsorptive (to lipid or protein domains), phagocytic and pinocytotic endocytosis, as well as passive diffusion. Localisation models discussed include those for cytosol, endoplasmic reticulum, Golgi apparatus, lipid droplets, lysosomes, mitochondria, nucleus and plasma membrane. A case example illustrates how such QSAR modelling of probe interactions can clarify localisation and mode of binding of probes to intracellular nucleic acids of living cells, including not only eukaryotic chromatin DNA and ribosomal RNA, but also prokaryote chromosomes.     

Experiments on movement of DNA regions in Escherichia coli evaluated by computer simulation

During the cell cycle of Escherichia coli DNA is replicated and segregated over two prospective daughter cells. Nucleoids as a whole separate gradually in line with cell elongation, but sub-nucleoid DNA regions may behave differently, separating non-gradually. We tested the ability of three models to predict the outcome of a fluorescent in situ hybridisation (FISH) experiment. We did this by comparing computer-simulated data with experimental data. The first model predicts gradual separation in line with cell elongation. The second model predicts that origins stick together for some time after duplication before one copy jumps to the other side of the cell (non-gradual separation). The simulated data of these models are very similar, indicating that FISH is not a suitable method to distinguish between these two models. The third model predicts that origins may be anywhere within the nucleoid(s). We found that simulated data using the third model resemble the experimental data most. However, DNA regions are not randomly localised in the cell, although their localisation is fuzzy. We propose that movement of DNA regions is the result of a combination of factors. Nucleoid segregation (or the forces behind it) dictates the overall direction of movement. Other factors, of which we show that diffusion could be an important one, move DNA in other directions giving rise to non-gradual movement in individual cells and contributing to variation in intracellular position per cell length in a population of cells.