A distributed parameter identification problem in neuronal cable theory models

Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.     

Improving parameter estimation for cell surface FRAP data

Fluorescence Recovery After Photobleaching (FRAP) using the confocal laser scanning microscope has become a standard method used to determine the diffusion coefficient and mobile fraction of cell surface proteins. A common experimental approach is to bleach a stripe on the cell surface and fit the ensuing FRAP curve to a 1D diffusion model. This model is derived from the time course of recovery to an infinitely long stripe bleached on an infinite flat plane. This choice of model dictates the use of a long bleach stripe. We demonstrate that, in the case of a long bleach stripe, the finite extent of the cell leads to significant errors in parameter estimation. We further show that these errors are reduced when a relatively small stripe is bleached. Unfortunately, diffusion to such a region is fundamentally two dimensional and therefore applying the 1D model of diffusion leads to significant errors. We derive an equation suitable for fitting to FRAP data acquired from small bleach regions and analyze its accuracy using simulated data. We propose that the use of a small bleach region along with a two dimensional diffusion model is the ideal protocol for cell surface FRAP.     

Protein Sliding along DNA: Dynamics and Structural Characterization

Efficient search of DNA by proteins is fundamental to the control of cellular regulatory processes. It is currently believed that protein sliding, hopping, and transfer between adjacent DNA segments, during which the protein nonspecifically interacts with DNA, are central to the speed of their specific recognition. In this study, we focused on the structural and dynamic features of proteins when they scan the DNA. Using a simple computational model that represents protein-DNA interactions by electrostatic forces, we identified that the protein makes use of identical binding interfaces for both nonspecific and specific DNA interactions. Accordingly, in its one-dimensional diffusion along the DNA, the protein is bound at the major groove and performs a helical motion, which is stochastic and driven by thermal diffusion. A microscopic structural insight into sliding from our model, which is governed by electrostatic forces, corroborates previous experimental studies suggesting that the active site of some regulatory proteins continually faces the interior of the DNA groove while sliding along sugar-phosphate rails. The diffusion coefficient of spiral motion along the major groove of the DNA is not affected by salt concentration, but the efficiency of the search can be significantly enhanced by increasing salt concentration due to a larger number of hopping events. We found that the most efficient search comprises ∼ 20% sliding along the DNA and ∼ 80% hopping and three-dimensional diffusion. The presented model that captures various experimental features of facilitated diffusion has the potency to address other questions regarding the nature of DNA search, such as the sliding characteristics of oligomeric and multidomain DNA-binding proteins that are ubiquitous in the cell.     

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

A Re-Evaluation of the Role of the Infected Cell in the Control of O2 Diffusion in Legume Nodules

Two different simulation models were constructed to describe O2 diffusion into the bacteria-infected cells of legume nodules: one based on a central zone of uniform spherical cells and the other on a central zone of packed, uniform cubical cells with air spaces along the edges. The cubical model more closely approximated the geometry and gas diffusion characteristics of infected cells than did the spherical model. The models relied on set values for the innermost O2 concentration in the infected cell (1-20 nM) and predicted values for the free O2 and oxygenated leghemoglobin gradients toward the cell:space interface. The cubical model but not the spherical model predicted saturation of leghemoglobin (Lb) oxygenation at or within a few micrometers of the gas-filled intercellular space and predicted that the space concentration could be as high as 1.3% O2 when the fractional oxygenation of Lb and respiration rate within the infected cell were typical of that which has been measured in vivo. In the model, the higher the space O2 concentration, the greater the saturation of Lb by O2 and the greater the collapse of Lb-facilitated diffusion near the cell:space interface. This was predicted to result in a greater resistance to O2 diffusion from the space to the bacteroids, thereby providing an intrinsic, homeostatic mechanism for controlling the rate of O2 influx into infected cells. Changes in the physiological features of the simulated cubical infected cell, such as the proportion of the cell as cytosol, the surface area of the cell exposed to a space, the maximum rate of cellular respiration, or the concentration of Lb in the cytoplasm, significantly altered the extent to which the infected cell would be able to regulate its diffusive resistance. These results demonstrate the possibility of a Lb-based mechanism for controlling the O2 concentration within the infected cells. If such a mechanism exists in legume nodules, it would give the infected cell an ability to exercise fine control over its internal environment, a process that could complement a physical diffusion barrier that may exist in the inner cortex or elsewhere in the nodule and provide coarse control over O2 diffusion.     

Calcium oscillations in a triplet of pancreatic acinar cells

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.     

Mathematical model of the spatio-temporal dynamics of second messengers in visual transduction

A model describing the role of transversal and longitudinal diffusion of cGMP and Ca(2+) in signaling in the rod outer segment of vertebrates is developed. Utilizing a novel notion of surface-volume reaction and the mathematical theories of homogenization and concentrated capacity, the diffusion of cGMP and Ca(2+) in the inter-disc spaces is shown to be reducible to a one-parameter family of diffusion processes taking place on a single rod cross section; whereas the diffusion in the outer shell is shown to be reducible to a diffusion on a cylindrical surface. Moreover, the exterior flux of the former serves as a source term for the latter, alleviating the assumption of a well-stirred cytosol. A previous model of visual transduction that assumes a well-stirred rod outer segment cytosol (and thus contains no spatial information) can be recovered from this model by imposing a "bulk" assumption. The model shows that upon activation of a single rhodopsin, cGMP changes are local, and exhibit both a longitudinal and a transversal component. Consequently, membrane current is also highly localized. The spatial spread of the single photon response along the longitudinal axis of the outer segment is predicted to be 3-5 microm, consistent with experimental data. This approach represents a tool to analyze point-wise signaling dynamics without requiring averaging over the entire cell by global Michaelis-Menten kinetics.     

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.     

Mathematical model of activation wave propagation in an isolated cardiomyocyte

In order to describe spontaneous wave-like contractions of a single isolated cardiomyocyte a mathematical model is proposed, which relates this phenomenon to propagation of calcium ion concentration wave along the cell. Free diffusion of Ca2+ ions as well as their reversible binding to regulatory proteins in contractile apparatus, Ca2+ accumulation in sarcoplasmic reticulum, and Ca-induced Ca2+ release are included in the governing equations. The model agrees with some observations. It predicts also some effects which may by a subject of future experimental research.     

Coupled water transport in standing gradient models of the lateral intercellular space.

A standing gradient model of the lateral intercellular space is presented which includes a basement membrane of finite solute permeability. The solution to the model equations is estimated analytically using the "isotonic convection approximation" of Segel. In the case of solute pumps uniformly distributed along the length of the channel, the achievement of isotonic transport depends only on the water permeability of the cell membranes. The ability of the model to transport water against an adverse osmotic gradient is the sum of two terms: The first term is simply that for a well-stirred compartment model and reflects basement membrane solute permeability. The second term measures the added strength due to diffusion limitation within the interspace. It is observed, however, that the ability for uphill water transport due to diffusion limitation is diminished by high cell membrane water permeability. For physiologically relevant parameters, it appears that the high water permeability required for isotonic transport renders the contribution of the standing gradient relatively ineffective in transport against an osmotic gradient. Finally, when the model transports both isotonically and against a gradient, it is shown that substantial intraepithelial solute polarization effects are unavoidable. Thus, the measured epithelial water permeability will grossly underestimate the water permeability of the cell membranes. The accuracy of the analytic approximation is demonstrated by numerical solution of the complete model equations.     

Mathematical modeling of regulatory mechanisms in yeast colony development

In the present study, yeast colony development serves as a model system to study growth of fungal populations with negligible nutrient and signal transport within the mycelium. Mathematical simulations address the question whether colony development is governed by diffusional limitation of nutrients. A hybrid one-dimensional cellular automaton model was developed that describes growth of discrete cells based upon microscopic interaction rules in a continuous field of nutrient and messenger. The model is scaled for the geometry of the experimental setup, cell size, growth- and substrate uptake rates. Therefore, calculated cell density profiles and nutrient distributions can be compared to experimental results and the model assumptions can be verified. In the physiologically relevant parameter range, simulations show an exponentially declining cell density along the median axis of the colonies in case of a diffusion limited growth scenario. These results are in good agreement with cell density profiles obtained in cultivations of the yeast Candida boidinii with glucose as the limiting carbon source but stand in contrast to the constant cell density profile estimated for Yarrowia lipolytica grown under the same conditions. While from the comparison of experimental results and simulations a diffusion limited growth mechanism is proposed for glucose limited C. boidinii colonies, this hypothesis is rejected for the growth of Y. lipolytica. As an alternative, a quorum sensing model was developed that can explain the evolution of constant cell density profiles based on the effect of a not further characterized unstable or volatile messenger.     

The Kinetics of Vitamin D3 in the Osteoblastic Cell

Experimental evidence is presented on the translocation of vitamin D metabolite, 1,25-(OH)₂D₃, from the membrane to the nucleus in osteoblast progenitor cells. A mathematical model permitting traversal of the cytoplasm at either a fixed velocity or by diffusion is formulated in order to determine whether transport along the cytoskeletal tracks is more consistent with the observed spatial-temporal distribution than diffusion, and it is so found. The model includes reactions in the nucleus involving D₃ to form other compounds, such as protegerin, and thus also makes predictions of the concentrations of these compounds in various regions of the cell.     

Dynamics of outgrowth in a continuum model of neurite elongation

Neurite outgrowth (dendrites and axons) should be a stable, but easily regulated process to enable a neuron to make its appropriate network connections during development. We explore the dynamics of outgrowth in a mathematical continuum model of neurite elongation. The model describes the construction of the internal microtubule cytoskeleton, which results from the production and transport of tubulin dimers and their assembly into microtubules at the growing neurite tip. Tubulin is assumed to be largely synthesised in the cell body from where it is transported by active mechanisms and by diffusion along the neurite. It is argued that this construction process is a fundamental limiting factor in neurite elongation. In the model, elongation is highly stable when tubulin transport is dominated by either active transport or diffusion, but oscillations in length may occur when both active transport and diffusion contribute. Autoregulation of tubulin production can eliminate these oscillations. In all cases a stable steady-state length is reached, provided there is intrinsic decay of tubulin. Small changes in growth parameters, such as the tubulin production rate, can lead to large changes in length. Thus cytoskeleton construction can be both stable and easily regulated, as seems necessary for neurite outgrowth during nervous system development. Action Editor: Upinder Bhalla     

Association kinetics with coupled diffusion: III. Ionic-strength dependence of the lac repressor-operator association

The repressor- operator association is treated in a model where the represser molecule can find its specific binding site the operator, on a large DNA chain by performing a one-dimensional diffusion along the chain. The ionic-strength depen deuce is calculated by introducing a screened electrostatic potential around the DNA chain and coupling the free diffusion of the represser in this potential to the proposed one-dimensional diffusion along the chain. The main influence on the association rate comes from the competitive binding of ions to the unspecific DNA sites. It is also demonstrated that during the time that the represser is bound in a global sense, the diffusion along the chain will be made up of a strictly one-dimensional motion over fairly short distances, interspersed with many local dissociations during which the repressor in essence is free in solution.     

Dispersal of motile bacteria from a plane layer.

The dispersal of an initially well-defined concentration of the motile bacterium Escherichia coli was measured under nonchemotactic conditions. The distribution of bacteria along a glass observation cell was measured by recording the intensity of light scattered by the organisms. For comparison, the diffusion of fluorescein was also measured by determining the distribution of fluorescence throughout the observation cell. The dispersal of bacteria from a plane layer, under nonchemotactic conditions, can be adequately described by the Gaussian solution of the diffusion equation.     

Red Cell Membrane Permeability Deduced from Bulk Diffusion Coefficients

The permeability coefficients of dog red cell membrane to tritiated water and to a series of[¹⁴C]amides have been deduced from bulk diffusion measurements through a "tissue" composed of packed red cells. Red cells were packed by centrifugation inside polyethylene tubing. The red cell column was pulsed at one end with radiolabeled solute and diffusion was allowed to proceed for several hours. The distribution of radioactivity along the red cell column was measured by sequential slicing and counting, and the diffusion coefficient was determined by a simple plotting technique, assuming a one-dimensional diffusional model. In order to derive the red cell membrane permeability coefficient from the bulk diffusion coefficient, the red cells were assumed to be packed in a regular manner approximating closely spaced parallelopipeds. The local steady-state diffusional flux was idealized as a one-dimensional intracellular pathway in parallel with a one-dimensional extracellular pathway with solute exchange occurring within the series pathway and between the pathways. The diffusion coefficients in the intracellular and extracellular pathways were estimated from bulk diffusion measurements through concentrated hemoglobin solutions and plasma, respectively; while the volume of the extracellular pathway was determined using radiolabeled sucrose. The membrane permeability coefficients were in satisfactory agreement with the data of Sha'afi, R. I., C. M. Gary-Bobo, and A. K. Solomon (1971. J. Gen. Physiol. 58:238) obtained by a rapid-reaction technique. The method is simple and particularly well suited for rapidly permeating solutes.     

Diffusion approximation of the stochastic process of microtubule assembly

Microtubules are protein polymers that guide intracellular motility. Stochastic switching of a microtubule between states of elongation, shortening, and pause is described in detail by the dynamic instability (DI) model. Recently we have described the dynamics of microtubules phenomenologically as generalized diffusion of their ends. Genesis of the diffusion dynamics and accuracy of diffusion model are studied in this work. It is shown that wandering of the end of a microtubule undergoing DI asymptotically approaches the Wiener diffusion process. Accuracy of the diffusion approximation is evaluated by comparing its predictions with results of simulation of DI. Stationary distributions of microtubule length and lifetime that are predicted by both models differ qualitatively between two cell types considered. However, predictions of the diffusion model are in each case practically identical to predictions of the DI model being also consistent with experimental data. The peculiar stochastic process of microtubule assembly thus converges at cell scale to a kind of widespread-in-nature diffusion process. This result is considered an example of qualitative change in dynamical properties in transition from the molecular to cellular level of biological organization. Additionally, it suggests employment of diffusion process theory in studying functions of microtubules in the cell.