The Impact of Elastic Deformations of the Extracellular Matrix on Cell Migration.

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the direction and speed of migration. We model a cell in 1D as a nucleus connected to a number of adhesion sites through elastic springs. The cell migrates by randomly updating the position of its adhesion sites. We start by investigating the case where the cell springs are constant, and then go on to assuming that they depend on the matrix stiffness, on matrices of both uniform stiffness as well as those with a stiffness gradient. We find that the assumption that cell springs depend on the substrate stiffness is necessary and sufficient for an efficient durotactic response. We compare simulations to recent experimental observations of human cancer cells exhibiting durotaxis, which show good qualitative agreement.

Impact of anticipation in dynamical systems.

Many animals, including humans, have predictive capabilities and, presumably, base their behavioral decisions-at least partially-upon an anticipated state of their environment. We explore a minimal version of this idea in the context of particles that interact according to a pairwise potential. Anticipation enters the picture by calculating the interparticle forces from linear extrapolations of the particle positions some time τ in the future. Simulations show that for intermediate values of τ, compared to a transient time scale defined by the potential and the initial conditions, the particles form rotating clusters in which the particles are arranged in a hexagonal pattern. Analysis of the system shows that anticipation induces energy dissipation and we show that the kinetic energy asymptotically decays as 1/t. Furthermore, we show that the angular momentum is not necessarily conserved for τ>0, and that asymmetries in the initial condition therefore can cause rotational movement. These results suggest that anticipation could play an important role in collective behavior, since it may induce pattern formation and stabilizes the dynamics of the system.

Pathway identification by network pruning in the metabolic network of Escherichia coli.

MOTIVATION: All metabolic networks contain metabolites, such as ATP and NAD, known as currency metabolites, which take part in many reactions. These are often removed in the study of these networks, but no consensus exists on what actually constitutes a currency metabolite, and it is also unclear how these highly connected nodes contribute to the global structure of the network. RESULTS: In this article, we analyse how the Escherichia coli metabolic network responds to pruning in the form of sequential removal of metabolites with highest degree. As expected this leads to network fragmentation, but the process by which it occurs suggests modularity and long-range correlations within the network. We find that the pruned networks contain longer paths than the random expectation, and that the paths that survive the pruning also exhibit a lower cost (number of involved metabolites) compared with random paths in the full metabolic network. Finally we confirm that paths detected by pruning overlap with known metabolic pathways. We conclude that pruning reveals functional pathways in metabolic networks, where currency metabolites may be seen as ingredients in a well-balanced soup in which main metabolic production lines are immersed. CONTACT: gerlee@nbi.dk SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Evolution of cell motility in an individual-based model of tumour growth.

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.

Gene divergence and pathway duplication in the metabolic network of yeast and digital organisms.

We have studied the metabolic gene-function network in yeast and digital organisms evolved in the artificial life platform Avida. The gene-function network is a bipartite network in which a link exists between a gene and a function (pathway) if that function depends on that gene, and can also be viewed as a decomposition of the more traditional functional gene networks, where two genes are linked if they share any function. We show that the gene-function network exhibits two distinct degree distributions: the gene degree distribution is scale-free while the pathway distribution is exponential. This is true for both yeast and digital organisms, which suggests that this is a general property of evolving systems, and we propose that the scale-free gene degree distribution is due to pathway duplication, i.e. the development of a new pathway where the original function is still retained. Pathway duplication would serve as preferential attachment for the genes, and the experiments with Avida revealed precisely this; genes involved in many pathways are more likely to increase their connectivity. Measuring the overlap between different pathways, in terms of the genes that constitute them, showed that pathway duplication also is a likely mechanism in yeast evolution. This analysis sheds new light on the evolution of genes and functionality, and suggests that function duplication could be an important mechanism in evolution.

Modelling evolutionary cell behaviour using neural networks: application to tumour growth.

In this paper, we present a modelling framework for cellular evolution that is based on the notion that a cell's behaviour is driven by interactions with other cells and its immediate environment. We equip each cell with a phenotype that determines its behaviour and implement a decision mechanism to allow evolution of this phenotype. This decision mechanism is modelled using feed-forward neural networks, which have been suggested as suitable models of cell signalling pathways. The environmental variables are presented as inputs to the network and result in a response that corresponds to the phenotype of the cell. The response of the network is determined by the network parameters, which are subject to mutations when the cells divide. This approach is versatile as there are no restrictions on what the input or output nodes represent, they can be chosen to represent any environmental variables and behaviours that are of importance to the cell population under consideration. This framework was implemented in an individual-based model of solid tumour growth in order to investigate the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. Our results show that the oxygen concentration affects the tumour at the morphological level, but more importantly has a direct impact on the evolutionary dynamics. When the supply of oxygen is limited we observe a faster divergence away from the initial genotype, a higher population diversity and faster evolution towards aggressive phenotypes. The implementation of this framework suggests that this approach is well suited for modelling systems where evolution plays an important role and where a changing environment exerts selection pressure on the evolving population.

The emergence of overlapping scale-free genetic architecture in digital organisms.

We have studied the evolution of genetic architecture in digital organisms and found that the gene overlap follows a scale-free distribution, which is commonly found in metabolic networks of many organisms. Our results show that the slope of the scale-free distribution depends on the mutation rate and that the gene development is driven by expansion of already existing genes, which is in direct correspondence to the preferential growth algorithm that gives rise to scale-free networks. To further validate our results we have constructed a simple model of gene development, which recapitulates the results from the evolutionary process and shows that the mutation rate affects the tendency of genes to cluster. In addition we could relate the slope of the scale-free distribution to the genetic complexity of the organisms and show that a high mutation rate gives rise to a more complex genetic architecture.

A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype.

We present a cellular automaton model of clonal evolution in cancer aimed at investigating the emergence of the glycolytic phenotype. In the model each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. This implies that cells might react differently to the environment and when space and nutrients are limited only the fittest cells will survive. With this model we have investigated the impact of the environment on the growth dynamics of the tumour. In particular, we have analysed the influence of the tissue oxygen concentration and extra-cellular matrix density on the dynamics of the model. We found that the environment influences both the growth and the evolutionary dynamics of the tumour. For low oxygen concentration we observe tumours with a fingered morphology, while increasing the matrix density gives rise to more compact tumours with wider fingers. The distribution of phenotypes in the tumour is also affected, and we observe that the glycolytic phenotype is most likely to emerge in a poorly oxygenated tissue with a high matrix density. Our results suggest that it is the combined effect of the oxygen concentration and matrix density that creates an environment where the glycolytic phenotype has a growth advantage and consequently is most likely to appear.

Stability analysis of a hybrid cellular automaton model of cell colony growth.

Cell colonies of bacteria, tumor cells, and fungi, under nutrient limited growth conditions, exhibit complex branched growth patterns. In order to investigate this phenomenon we present a simple hybrid cellular automaton model of cell colony growth. In the model the growth of the colony is limited by a nutrient that is consumed by the cells and which inhibits cell division if it falls below a certain threshold. Using this model we have investigated how the nutrient consumption rate of the cells affects the growth dynamics of the colony. We found that for low consumption rates the colony takes on an Eden-like morphology, while for higher consumption rates the morphology of the colony is branched with a fractal geometry. These findings are in agreement with previous results, but the simplicity of the model presented here allows for a linear stability analysis of the system. By observing that the local growth of the colony is proportional to the flux of the nutrient we derive an approximate dispersion relation for the growth of the colony interface. This dispersion relation shows that the stability of the growth depends on how far the nutrient penetrates into the colony. For low nutrient consumption rates the penetration distance is large, which stabilizes the growth, while for high consumption rates the penetration distance is small, which leads to unstable branched growth. When the penetration distance vanishes the dispersion relation is reduced to the one describing Laplacian growth without ultra-violet regularization. The dispersion relation was verified by measuring how the average branch width depends on the consumption rate of the cells and shows good agreement between theory and simulations.

An evolutionary hybrid cellular automaton model of solid tumour growth.

We propose a cellular automaton model of solid tumour growth, in which each cell is equipped with a micro-environment response network. This network is modelled using a feed-forward artificial neural network, that takes environmental variables as an input and from these determines the cellular behaviour as the output. The response of the network is determined by connection weights and thresholds in the network, which are subject to mutations when the cells divide. As both available space and nutrients are limited resources for the tumour, this gives rise to clonal evolution where only the fittest cells survive. Using this approach we have investigated the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. The results show that the oxygen concentration affects the selection pressure, cell population diversity and morphology of the tumour. A low oxygen concentration in the tissue gives rise to a tumour with a fingered morphology that contains aggressive phenotypes with a small apoptotic potential, while a high oxygen concentration in the tissue gives rise to a tumour with a round morphology containing less evolved phenotypes. The tissue oxygen concentration thus affects the tumour at both the morphological level and on the phenotype level.