Jeppe Juul

Equilibrium strategy and population-size effects in lowest unique bid auctions.

In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large data set of internet auctions. The theory agrees with the data with striking accuracy for small population-size N, while for larger N a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.

Labyrinthine clustering in a spatial rock-paper-scissors ecosystem.

The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.

Clonal selection prevents tragedy of the commons when neighbors compete in a rock-paper-scissors game

The rock-paper-scissors game is a model example of the ongoing cyclic turnover typical of many ecosystems, ranging from the terrestrial and aquatic to the microbial. Here we explore the evolution of a rock-paper-scissors system where three species compete for space. The species are allowed to mutate and change the speed by which they invade one another. In the case when all species have similar mutation rates, we observe a perpetual arms race where no single species prevails. When only two species mutate, their aggressions increase indefinitely until the ecosystem collapses and only the nonmutating species survives. Finally we show that when only one species mutates, group selection removes individual predators with the fastest growth rates, causing the growth rate of the species to stabilize. We explain this group selection quantitatively.

A kinetic mechanism for cell sorting based on local variations in cell motility.

Our current understanding of cell sorting relies on physical difference, either in the interfacial properties or motile force, between cell types. But is such asymmetry a prerequisite for cell sorting? We test this using a minimal model in which the two cell populations are identical with respect to their physical properties and differences in motility arise solely from how cells interact with their surroundings. The model resembles the Schelling model used in social sciences to study segregation phenomena at the scale of societies. Our results demonstrate that segregation can emerge solely from cell motility being a dynamic property that changes in response to the local environment of the cell, but that additional mechanisms are necessary to reproduce the envelopment behaviour observed in vitro. The time course of segregation follows a power law, in agreement with the scaling reported from experiment and in other models of motility-driven segregation.

Replicator dynamics with turnover of players.

We study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies. We demonstrate how such a process leads macroscopically to a variant of the replicator equation, with an additional term accounting for player turnover. We study how Nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock-paper-scissors game and different classes of two-action games played between two distinct populations. We conclude by showing how player turnover can account for nontrivial departures from Nash equilibria observed in data from lowest unique bid auctions.

Asymmetric segregation of damaged cellular components in spatially structured multicellular organisms.

The asymmetric distribution of damaged cellular components has been observed in species ranging from fission yeast to humans. To study the potential advantages of damage segregation, we have developed a mathematical model describing ageing mammalian tissue, that is, a multicellular system of somatic cells that do not rejuvenate at cell division. To illustrate the applicability of the model, we specifically consider damage incurred by mutations to mitochondrial DNA, which are thought to be implicated in the mammalian ageing process. We show analytically that the asymmetric distribution of damaged cellular components reduces the overall damage level and increases the longevity of the cell population. Motivated by the experimental reports of damage segregation in human embryonic stem cells, dividing symmetrically with respect to cell-fate, we extend the model to consider spatially structured systems of cells. Imposing spatial structure reduces, but does not eliminate, the advantage of asymmetric division over symmetric division. The results suggest that damage partitioning could be a common strategy for reducing the accumulation of damage in a wider range of cell types than previously thought.

Noisy transcription factor NF-κB oscillations stabilize and sensitize cytokine signaling in space.

NF-κB is a major transcription factor mediating inflammatory response. In response to a pro-inflammatory stimulus, it exhibits a characteristic response-a pulse followed by noisy oscillations in concentrations of considerably smaller amplitude. NF-κB is an important mediator of cellular communication, as it is both activated by and upregulates production of cytokines, signals used by white blood cells to find the source of inflammation. While the oscillatory dynamics of NF-κB has been extensively investigated both experimentally and theoretically, the role of the noise and the lower secondary amplitude has not been addressed. We use a cellular automaton model to address these issues in the context of spatially distributed communicating cells. We find that noisy secondary oscillations stabilize concentric wave patterns, thus improving signal quality. Furthermore, both lower secondary amplitude as well as noise in the oscillation period might be working against chronic inflammation, the state of self-sustained and stimulus-independent excitations. Our findings suggest that the characteristic irregular secondary oscillations of lower amplitude are not accidental. On the contrary, they might have evolved to increase robustness of the inflammatory response and the systems ability to return to a pre-stimulated state.

Schelling model of cell segregation based only on local information

While biological studies suggest that motility of cells is involved in cell segregation, few computational models have investigated this mechanism. We apply a simple Schelling model, modified to reflect biological conditions, demonstrating how differences in cell motility arising exclusively from differences in the composition of the local environment can be sufficient to drive segregation. The work presented here demonstrates that the segregation behavior observed in the original Schelling model is robust to a relaxation of the requirement for global information and that the Schelling model may yield insight in the context of biological systems. In the model, the time course of cell segregation follows a power law in accord with experimental observations and previous work.

Moderate stem-cell telomere shortening rate postpones cancer onset in a stochastic model.

Mammalian cells are restricted from proliferating indefinitely. Telomeres at the end of each chromosome are shortened at cell division and when they reach a critical length, the cell will enter permanent cell cycle arrest-a state known as senescence. This mechanism is thought to be tumor suppressing, as it helps prevent precancerous cells from dividing uncontrollably. Stem cells express the enzyme telomerase, which elongates the telomeres, thereby postponing senescence. However, unlike germ cells and most types of cancer cells, stem cells only express telomerase at levels insufficient to fully maintain the length of their telomeres, leading to a slow decline in proliferation potential. It is not yet fully understood how this decline influences the risk of cancer and the longevity of the organism. We here develop a stochastic model to explore the role of telomere dynamics in relation to both senescence and cancer. The model describes the accumulation of cancerous mutations in a multicellular organism and creates a coherent theoretical framework for interpreting the results of several recent experiments on telomerase regulation. We demonstrate that the longest average cancer-free lifespan before cancer onset is obtained when stem cells start with relatively long telomeres that are shortened at a steady rate at cell division. Furthermore, the risk of cancer early in life can be reduced by having a short initial telomere length. Finally, our model suggests that evolution will favor a shorter than optimal average cancer-free lifespan in order to postpone cancer onset until late in life.

Fragile DNA Repair Mechanism Reduces Ageing in Multicellular Model

DNA damages, as well as mutations, increase with age. It is believed that these result from increased genotoxic stress and decreased capacity for DNA repair. The two causes are not independent, DNA damage can, for example, through mutations, compromise the capacity for DNA repair, which in turn increases the amount of unrepaired DNA damage. Despite this vicious circle, we ask, can cells maintain a high DNA repair capacity for some time or is repair capacity bound to continuously decline with age? We here present a simple mathematical model for ageing in multicellular systems where cells subjected to DNA damage can undergo full repair, go apoptotic, or accumulate mutations thus reducing DNA repair capacity. Our model predicts that at the tissue level repair rate does not continuously decline with age, but instead has a characteristic extended period of high and non-declining DNA repair capacity, followed by a rapid decline. Furthermore, the time of high functionality increases, and consequently slows down the ageing process, if the DNA repair mechanism itself is vulnerable to DNA damages. Although counterintuitive at first glance, a fragile repair mechanism allows for a faster removal of compromised cells, thus freeing the space for healthy peers. This finding might be a first step toward understanding why a mutation in single DNA repair protein (e.g. Wrn or Blm) is not buffered by other repair proteins and therefore, leads to severe ageing disorders.