Cell orientation response to cyclically deformed substrates: Experimental validation of a cell model

We have developed a stochastic model that describes the orientation response of bipolar cells grown on a cyclically deformed substrate. The model was based on the following hypotheses regarding the behavior of individual cells: (a) the mechanical signal responsible for cell reorientation is the peak to peak surface strain along the cell's major axis (p-p axial strain); (b) each cell has an axial strain threshold and the threshold is normally distributed in the cell population; (c) the cell will avoid any direction where the p-p axial strain is above its threshold; and (d) the cell will randomly orient within the range of directions where the p-p axial strains are less than the cell's threshold. These hypotheses were tested by comparing model predictions with experimental observations from stretch experiments conducted with human melanocytes. The cells were grown on elastic rectangular culture dishes subjected to unidirectional cyclic (1 Hz) stretching with amplitudes of 0, 4, 8, and 12%. After 24 h of stimulation, the distribution of cell orientations was determined by measuring the orientations of 300–400 randomly selected cells. The 12% stretch experiment was used to determine the mean, 3.5%, and the standard deviation, 1.0%, of the strain threshold for the cell population. The Kolmogorov-Smirnov test was then used to determine if the orientation distributions predicted by the model were different from experimentally measured distributions for the 4 and 8% stretches. No significant differences were found between the predicted and experimental distributions (4%: p = 0.70; and 8%: p = 0.71). These results support the hypothesis that cells randomly orient, but avoid directions where the p-p axial strains are above their thresholds.     

Global cell sorting in the C. elegans embryo defines a new mechanism for pattern formation

4D microscopic observations of Caenorhabditis elegans development show that the nematode uses an unprecedented strategy for development. The embryo achieves pattern formation by sorting cells, through far-ranging movements, into coherent regions before morphogenesis is initiated. This sorting of cells is coupled to their particular fate. If cell identity is altered by experiment, cells are rerouted to positions appropriate to their new fates even across the whole embryo. This cell behavior defines a new mechanism of pattern formation, a mechanism that is also found in other animals. We call this new mechanism "cell focusing". When the fate of cells is changed, they move to new positions which also affect the shape of the body. Thus, this process is also important for morphogenesis.     

On a simplified model for pattern formation in honey bee colonies

We present a simplified version of a previously presented model (Camazine et al. (1990)) that generates the characteristic pattern of honey, pollen and brood which develops on combs in honey bee colonies. We demonstrate that the formation of a band of pollen surrounding the brood area is dependent on the assumed form of the honey and pollen removal terms, and that a significant pollen band arises as the parameter controlling the rate of pollen input passes through a bifurcation value. The persistence of the pollen band after a temporary increase in pollen input can be predicted from the model. We also determine conditions on the parameters which ensure the accumulation of honey in the periphery and demonstrate that, although there is an important qualitative difference between the simplified and complete models, an analysis of the simplified version helps us understand many biological aspects of the more complex complete model. Corresponding author     

Ameboid cell motility: a model and inverse problem, with an application to live cell imaging data

In this article a mathematical model for ameboid cell movement is developed using a spring-dashpot system with Newtonian dynamics. The model is based on the facts that the cytoskeleton plays a primary role for cell motility and that the cytoplasm is viscoelastic. Based on the model, the inverse problem can be posed: if a structure like a spring-dashpot system is embedded into the living cell, what kind of characteristic properties must the structure have in order to reproduce a given movement of the cell? This inverse problem is the primary topic of this paper. On one side the model mimics some features of the movement, and on the other side, the solution to the inverse problem provides model parameters that give some insight, principally into the mechanical aspect, but also, through qualitative reasoning, into chemical and biophysical aspects of the cell. Moreover, this analysis can be done locally or globally and in different media by using the simplest possible information: positions of the cell and nuclear membranes. It is shown that the model and solution to the inverse problem for simulated data sets are highly accurate. An application to a set of live cell imaging data obtained from random movements of a human brain tumor cell (U87-MG human glioblastoma cell line) then provides an example of the efficiency of the model, through the solution of its inverse problem, as a way of understanding experimental data.     

Cyclic AMP induces the synthesis of developmentally regulated plasma membrane proteins in Dictyostelium

Dictyostelium discoideum slugs (pseudoplasmodia) were disaggregated and the resynthesis of developmentally regulated plasma membrane proteins examined. The synthesis of the majority of these proteins was inhibited when cells were overlaid with Cellophane and maintained as a monolayer. However, cell contact and movement did occur under the Cellophane. The inhibition of differentiation may result from the inability of the cells to organise specifically into multicellular aggregates. The addition of cyclic AMP (1–5 mM) induced the synthesis of certain developmentally regulated plasma membrane proteins in cells overlaid with Cellophane. Hence, this confirms other work showing that cyclic AMP is required for at least some post-aggregative gene expression. Specific cell organisation and interactions are apparently required for an increase in or maintenance of intracellular cyclic AMP levels.     

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.     

The effects of coating culture dishes with collagen on fibroblast cell shape and swirling pattern formation

Motile human-skin fibroblasts form macroscopic swirling patterns when grown to confluence on a culture dish. In this paper, we investigate the effect of coating the culture-dish surface with collagen on the resulting pattern, using human-skin fibroblast NB1RGB cells as the model system. The presence of the collagen coating is expected to enhance the adherence of the fibroblasts to the dish surface, and thereby also enhance the traction that the fibroblasts have as they move. We find that, contrary to our initial expectation, the coating does not significantly affect the motility of the fibroblasts. Their eventual number density at confluence is also unaffected. However, the coherence length of cell orientation in the swirling pattern is diminished. We also find that the fibroblasts cultured in collagen-coated dishes are rounder in shape and shorter in perimeter, compared with those cultured in uncoated polystyrene or glass culture dishes. We hypothesise that the rounder cell-shape which weakens the cell–cell nematic contact interaction is responsible for the change in coherence length. A simple mathematical model of the migrating fibroblasts is constructed, which demonstrates that constant motility with weaker nematic interaction strength does indeed lead to the shortening of the coherence length.Electronic supplementary materialThe online version of this article (10.1007/s10867-020-09556-3) contains supplementary material, which is available to authorized users.     

A model for budding in hydra: pattern formation in concentric rings

Current models of pattern formation in Hydra propose head-and foot-specific morphogens to control the development of the body ends and along the body length axis. In addition, these morphogens are proposed to control a cellular parameter (positional value, source density) which changes gradually along the axis. This gradient determines the tissue polarity and the regional capacity to form a head and a foot, respectively, in transplantation experiments. The current models are very successful in explaining regeneration and transplantation experiments. However, some results obtained render problems, in particular budding, the asexual way of reproduction is not understood. Here an alternative model is presented to overcome these problems. A primary system of interactions controls the positional values. At certain positional values secondary systems become active which initiate the local formation of e.g. mouth, tentacles, and basal disc. (i) A system of autocatalysis and lateral inhibition is suggested to exist as proposed by Gierer and Meinhardt (Kybernetik 12 (1972) 30). (ii) The activator is neither a head nor a foot activator but rather causes an increase of the positional value. (iii) On the other hand, a generation of the activator leads to its loss from cells and therewith to a (local) decrease of the positional value. (iv) An inhibitor is proposed to exist which antagonizes an increase of the positional value. External conditions like the gradient of positional values in the surroundings and interactions with other sites of morphogen production decide whether at a certain site of activator generation the positional value will increase (head formation), decrease (foot formation) or increase in the centre and decrease in the periphery thereby forming concentric rings (bud formation). Computer-simulation experiments show basic features of budding, regeneration and transplantation.     

Using a stem cell and progeny model to illustrate the relationship between cell cycle times of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them

There is increasing evidence that the growth of human tumours is driven by a small proportion of tumour stem cells with self-renewal properties. Multiplication of these cells leads to loss of self-renewal and after division for a finite number of times the cells undergo programmed cell death. Cell cycle times of human cancers have been measured in vivo and shown to vary in the range from two days to several weeks, depending on the individual. Cells cultured directly from tumours removed at surgery initially grow at a rate comparable to the in vivo rate but continued culture leads to the generation of cell lines that have shorter cycle times (1–3 days). It has been postulated that the more rapidly growing sub-population exhibits some of the properties of tumour stem cells and are the precursors of a slower growing sub-population that comprise the bulk of the tumour. We have previously developed a mathematical model to describe the behaviour of cell lines and we extend this model here to describe the behaviour of a system with two cell populations with different kinetic characteristics and a precursor–product relationship. The aim is to provide a framework for understanding the behaviour of cancer tissue that is sustained by a minor population of proliferating stem cells.     

Cell spreading and motility: A model lamellipod

Cells moving in vitro do so by means of a motile appendage, the lamellipod. This is a broad, flat sheet of cytogel which spreads in front of the cell and pulls the cell forward. We present here a mathematical model for lamellipodial motion based on the physical chemistry of actomyosin gels.     

Stochastic Petri Net extension of a yeast cell cycle model

This paper presents the definition, solution and validation of a stochastic model of the budding yeast cell cycle, based on Stochastic Petri Nets (SPN). A specific family of SPNs is selected for building a stochastic version of a well-established deterministic model. We describe the procedure followed in defining the SPN model from the deterministic ODE model, a procedure that can be largely automated. The validation of the SPN model is conducted with respect to both the results provided by the deterministic one and the experimental results available from literature. The SPN model catches the behavior of the wild type budding yeast cells and a variety of mutants. We show that the stochastic model matches some characteristics of budding yeast cells that cannot be found with the deterministic model. The SPN model fine-tunes the simulation results, enriching the breadth and the quality of its outcome.     

Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs

Mathematic models help interpret experimental results and accelerate tissue engineering developments. We develop in this paper a hybrid cellular automata model that combines the differential nutrient transport equation to investigate the nutrient limited cell construct development for cartilage tissue engineering. Individual cell behaviors of migration, contact inhibition and cell collision, coupled with the cell proliferation regulated by oxygen concentration were carefully studied. Simplified two-dimensional simulations were performed. Using this model, we investigated the influence of cell migration speed on the overall cell growth within in vitro cell scaffolds. It was found that intense cell motility can enhance initial cell growth rates. However, since cell growth is also significantly modulated by the nutrient contents, intense cell motility with conventional uniform cell seeding method may lead to declined cell growth in the final time because concentrated cell population has been growing around the scaffold periphery to block the nutrient transport from outside culture media. Therefore, homogeneous cell seeding may not be a good way of gaining large and uniform cell densities for the final results. We then compared cell growth in scaffolds with various seeding modes, and proposed a seeding mode with cells initially residing in the middle area of the scaffold that may efficiently reduce the nutrient blockage and result in a better cell amount and uniform cell distribution for tissue engineering construct developments.     

Continuous Models for Cell Migration in Tissues and Applications to Cell Sorting via Differential Chemotaxis

Chemotaxis, the guided migration of cells in response to chemical gradients, is vital to a wide variety of biological processes, including patterning of the slime mold Dictyostelium, embryonic morphogenesis, wound healing, and tumor invasion. Continuous models of chemotaxis have been developed to describe many such systems, yet few have considered the movements within a heterogeneous tissue composed of multiple subpopulations. In this paper, a partial differential equation (PDE) model is developed to describe a tissue formed from two distinct chemotactic populations. For a “crowded” (negligible extracellular space) tissue, it is demonstrated that the model reduces to a simpler one-species system while for an “uncrowded” tissue, it captures both movement of the entire tissue (via cells attaching to/migrating within an extracellular substrate) and the within-tissue rearrangements of the separate cellular subpopulations. The model is applied to explore the sorting of a heterogeneous tissue, where it is shown that differential-chemotaxis not only generates classical sorting patterns previously seen via differential-adhesion, but also demonstrates new classes of behavior. These new phenomena include temporal dynamics consisting of a traveling wave composed of spatially sorted subpopulations reminiscent of Dictyostelium slugs.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.