Steady state distributions of moving particles in one dimension: with an eye towards axonal transport.

Axonal transport, propelled by motor proteins, plays a crucial role in maintaining the homeostasis of functional and structural components over time. To establish a steady-state distribution of moving particles, what conditions are necessary for axonal transport? This question is pertinent, for instance, to both neurofilaments and mitochondria, which are structural and functional cargoes of axonal transport. In this paper we prove four theorems regarding steady state distributions of moving particles in one dimension on a finite domain.

Mathematical model for IL-2-based cancer immunotherapy.

A basic mathematical model for IL-2-based cancer immunotherapy is proposed and studied. Our analysis shows that the outcome of therapy is mainly determined by three parameters, the relative death rate of CD4 T cells, the relative death rate of CD8 T cells, and the dose of IL-2 treatment. Minimal equilibrium tumor size can be reached with a large dose of IL-2 in the case that CD4 T cells die out.

Differential cell motion: A mathematical model of anterior posterior sorting.

Here, we investigate how a subpopulation of cells can move through an aggregate of cells. Using a stochastic force-based model of Dictyostelium discoideum when the population is forming a slug, we simulate different strategies for prestalk cells to reliably move to the front of the slug while omitting interaction with the substrate thus ignoring the overall motion of the slug. Of the mechanisms that we simulated, prestalk cells being more directed is the best strategy followed by increased asymmetric motive forces for prestalk cells.

Using Fluorescence Recovery After Photobleaching data to uncover filament dynamics.

Fluorescence Recovery After Photobleaching (FRAP) has been extensively used to understand molecular dynamics in cells. This technique when applied to soluble, globular molecules driven by diffusion is easily interpreted and well understood. However, the classical methods of analysis cannot be applied to anisotropic structures subjected to directed transport, such as cytoskeletal filaments or elongated organelles transported along microtubule tracks.

Utilizing Nonequilibrium Isotope Enrichments to Dramatically Increase Turnover Measurement Ranges in Single Biopsy Samples from Humans.

The synthesis of new proteins and the degradation of old proteins in vivo can be quantified in serial samples using metabolic isotope labeling to measure turnover. Because serial biopsies in humans are impractical, we set out to develop a method to calculate the turnover rates of proteins from single human biopsies. This method involved a new metabolic labeling approach and adjustments to the calculations used in previous work to calculate protein turnover.

Impact of noise on the regulation of intracellular transport of intermediate filaments.

Noise affects all biological processes from molecules to cells, organisms and populations. Although the effect of noise on these processes is highly variable, evidence is accumulating which shows natural stochastic fluctuations (noise) can facilitate biological functions. Herein, we investigate the effect of noise on the transport of intermediate filaments in cells by comparing the stochastic and deterministic formalizations of the bidirectional transport of intermediate filaments, long elastic polymers transported along microtubules by antagonistic motor proteins (Dallon et al.

A mathematical analysis of focal adhesion lifetimes and their effect on cell motility.

By analyzing the distributions of focal adhesion (FA) lifetimes from different cell types, we found that a gamma distribution best matched the experimental distributions. In all but one case, it was a unimodal, non-symmetric gamma distribution. We used a mathematical model of cell motion to help understand the mechanics and data behind the FA lifetime distributions.

Mean square displacement for a discrete centroid model of cell motion.

The mean square displacement (MSD) is an important statistical measure on a stochastic process or a trajectory. In this paper we find an approximation to the mean square displacement for a model of cell motion. The model is a discrete-time jump process which approximates a force-based model for cell motion.

How Structural Features of a Spring-Based Model of Fibrous Collagen Tissue Govern the Overall Young's Modulus.

While much study has been dedicated to investigating biopolymers' stress-strain response at low strain levels, little research has been done to investigate the almost linear region of biopolymers' stress-strain response and how the microstructure affects it. We propose a mathematical model of fibrous networks, which reproduces qualitative features of collagen gel's stress-strain response and provides insight into the key features which impact the Young's modulus of similar fibrous tissues. This model analyzes the relationship of the Young's modulus of the lattice to internodal fiber length, number of connection points or nodes per unit area, and average number of connections to each node.

Deciphering the transport of elastic filaments by antagonistic motor proteins.

Intermediate filaments are long elastic fibers that are transported by the microtubule-associated motor proteins kinesin and dynein inside the cell. How elastic filaments are efficiently transported by antagonistic motors is not well understood and is difficult to measure with current experimental techniques. Adapting the tug-of-war paradigm for vesiclelike cargos, we develop a mathematical model to describe the motion of an elastic filament punctually bound to antagonistic motors.

Stochastic modeling reveals how motor protein and filament properties affect intermediate filament transport.

Intermediate filaments are a key component of the cytoskeleton. Their transport along microtubules plays an essential role in the control of the shape and structural organization of cells. To identify the key parameters responsible for the control of intermediate filament transport, we generated a model of elastic filament transport by microtubule-associated dynein and kinesin.

A continuous-time model of centrally coordinated motion with random switching.

This paper considers differential problems with random switching, with specific applications to the motion of cells and centrally coordinated motion. Starting with a differential-equation model of cell motion that was proposed previously, we set the relaxation time to zero and consider the simpler model that results. We prove that this model is well-posed, in the sense that it corresponds to a pure jump-type continuous-time Markov process (without explosion).

A mathematical model of collagen lattice contraction.

Two mathematical models for fibroblast-collagen interaction are proposed which reproduce qualitative features of fibroblast-populated collagen lattice contraction. Both models are force based and model the cells as individual entities with discrete attachment sites; however, the collagen lattice is modelled differently in each model. In the collagen lattice model, the lattice is more interconnected and formed by triangulating nodes to form the fibrous structure.

Cell speed is independent of force in a mathematical model of amoeboidal cell motion with random switching terms.

In this paper the motion of a single cell is modeled as a nucleus and multiple integrin based adhesion sites. Numerical simulations and analysis of the model indicate that when the stochastic nature of the adhesion sites is a memoryless and force independent random process, the cell speed is independent of the force these adhesion sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment of the adhesion sites is key to predicting cell speed.

A force based model of individual cell migration with discrete attachment sites and random switching terms.

A force based model of cell migration is presented which gives new insight into the importance of the dynamics of cell binding to the substrate. The main features of the model are the focus on discrete attachment dynamics, the treatment of the cellular forces as springs, and an incorporation of the stochastic nature of the attachment sites. One goal of the model is to capture the effect of the random binding and unbinding of cell attachments on global cell motion.

Understanding streaming in Dictyostelium discoideum: theory versus experiments.

Recent experimental work involving Dictyostelium discoideum seems to contradict several theoretical models. Experiments suggest that localization of the release of the chemoattractant cyclic adenosine monophosphate to the uropod of the cell is important for stream formation during aggregation. Yet several mathematical models are able to reproduce streaming as the cells aggregate without taking into account localization of the chemoattractant.

Differences in the mechanism of collagen lattice contraction by myofibroblasts and smooth muscle cells.

Both rat derived vascular smooth muscle cells (SMC) and human myofibroblasts contain α smooth muscle actin (SMA), but they utilize different mechanisms to contract populated collagen lattices (PCLs). The difference is in how the cells generate the force that contracts the lattices. Human dermal fibroblasts transform into myofibroblasts, expressing α-SMA within stress fibers, when cultured in lattices that remain attached to the surface of a tissue culture dish.

Using a mathematical model of cadherin-based adhesion to understand the function of the actin cytoskeleton.

The actin cytoskeleton plays a role in cell-cell adhesion but its specific function is not clear. Actin might anchor cadherins or drive membrane protrusions in order to facilitate cell-cell adhesion. Using a mathematical model of the forces involved in cadherin-based adhesion, we investigate its possible functions.

A review of fibroblast-populated collagen lattices.

Bell's introduction of the fibroblast-populated collagen lattice (FPCL) has facilitated the study of collagen-cell interactions. As a result of the numerous modifications of the casting of FPCLs, the in vivo applications of these in vitro findings have been confusing. Here experimental FPCL contraction findings are viewed in regard to three proposed mechanisms responsible for lattice contraction.

Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications.

The extent to which collagen alignment occurs during dermal wound healing determines the severity of scar tissue formation. We have modelled this using a multiscale approach, in which extracellular materials, for example collagen and fibrin, are modelled as continua, while fibroblasts are considered as discrete units. Within this model framework, we have explored the effects that different parameters have on the alignment process, and we have used the model to investigate how manipulation of transforming growth factor-beta levels can reduce scar tissue formation.

Mathematically modelling the effects of counting factor in Dictyostelium discoideum.

Size regulation is a crucial feature in many biological systems, with misregulation leading to dysplasia or hyperplasia. The recent discovery of counting factor (CF) in Dictyostelium discoideum will lead to a greater understanding of how the system regulates the size of a group of cells. In this paper we mathematically model the known effects of CF using two different models: a cellular automata model and a discrete continuum hybrid model.

How cellular movement determines the collective force generated by the Dictyostelium discoideum slug.

How the collective motion of cells in a biological tissue originates in the behavior of a collection of individuals, each of which responds to the chemical and mechanical signals it receives from neighbors, is still poorly understood. Here we study this question for a particular system, the slug stage of the cellular slime mold Dictyostelium discoideum (Dd). We investigate how cells in the interior of a migrating slug can effectively transmit stress to the substrate and thereby contribute to the overall motive force.

Theoretical models of wound healing: past successes and future challenges.

The complex biology of wound healing is an area in which theoretical modelling has already made a significant impact. In this review article, the authors describe the key features of wound healing biology, divided into four components: epidermal wound healing, remodelling of the dermal extracellular matrix, wound contraction, and angiogenesis. Within each of these categories, previous modelling work is described, and the authors identify what they regard as the main challenges for future theoretical work.

Modeling the effects of transforming growth factor-beta on extracellular matrix alignment in dermal wound repair.

We present a novel mathematical model for collagen deposition and alignment during dermal wound healing, focusing on the regulatory effects of transforming growth factor-beta (TGFbeta.) Our work extends a previously developed model which considers the interactions between fibroblasts and an extracellular matrix composed of collagen and a fibrin based blood clot, by allowing fibroblasts to orient the collagen matrix, and produce and degrade the extracellular matrix, while the matrix directs the fibroblasts and control their speed. Here we extend the model by allowing a time varying concentration of TGFbeta to alter the properties of the fibroblasts.

Biological implications of a discrete mathematical model for collagen deposition and alignment in dermal wound repair.

We develop a novel mathematical model for collagen deposition and alignment during dermal wound healing. We focus on the interactions between fibroblasts, modelled as discrete entities, and a continuous extracellular matrix composed of collagen and a fibrin based blood clot. There are four basic interactions assumed in the model: fibroblasts orient the collagen matrix, fibroblasts produce and degrade collagen and fibrin and the matrix directs the fibroblasts and determines the speed of the cells.