## PhD Thesis: Spatial models and networks of living systems

When studying the dynamics of living systems, insight can often be gained by deriving a mathematical model that can predict future behaviour of the system or help classify the system characteristics. However, in living cells, organisms, and especially groups of interacting individuals, a large number of different factors influence the time development of the system. This often makes it challenging to construct a mathematical model from which one can draw conclusions.

One traditional way of capturing the dynamics in a mathematical model is to formulate a set of coupled differential equations for the essential variables of the system. However, this approach disregards any spatial structure of the system, which may potentially change the behaviour drastically. An alternative approach is to construct a cellular automaton with nearest neighbour interactions, or even to model the system as a complex network with interactions defined by network topology.

The main part of the thesis is based on six different articles, which I have co-authored during my three year PhD at the Center for Models of Life. Apart from these, I have co-authored another six articles, which also relate to spatial models of living systems. These are included as appendixes, but not described in detail in the thesis.

The thesis can be downloaded here.

## Replicator dynamics with turnover of players

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*By Jeppe Juul, Ardeshir Kianercy, Sebastian Bernhardsson, and Simone Pigolotti.*

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 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-scissor-paper game and different classes of two-action games played between two distinct populations. We conclude by showing how player turnover can account for non-trivial departures from Nash equilibria observed in data from lowest unique bid auctions.

The article can be found here.

## Asymmetric damage segregation in spatially structured multicellular organisms

By Charlotte Strandkvist, Jeppe Juul, and Kristian Moss Bendtsen

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 partitioning, we have developed a mathematical model describing a system of mammalian cells that duplicate damage during cell division. In particular, we consider defective mitochondria, which are thought to be a major contributor to the cellular deterioration associated with ageing. We show analytically that the asymmetric distribution of damage reduces the overall damage level of the population and delays the onset of clonal senescence. Motivated by the experimental reports of damage segregation in human embryonic stem cells, 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 range of cell types.

The article can be found here.

## Noisy NF-kB oscillations stabilize and sensitize cytokine signaling in space

By Sirin W. Gangstad, Cilie W. Feldager, Jeppe Juul, and Ala Trusina.

NF-kB is a major transcription factor mediating inflammatory response. In response to pro-inflammatory stimulus, it exhibits characteristic response — a pulse followed by noisy oscillations in concentrations of considerably smaller amplitude.NF-kB 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-kB 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 system’s ability to return to pre-stimulated state.

The article can be found here.

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

By Jeppe Juul, Kim Sneppen, and Joachim Mathiesen

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 new 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.

The article can be found here.

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

By Jeppe Juul, Kim Sneppen, and Joachim Mathiesen

The rock-paper-scissors game is a model example of the on-going 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 non-mutating 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.

The article can be found here.

## Fragile DNA Repair Mechanism Reduces Ageing in Multicellular Model

By Kristian Moss Bendtsen, Jeppe Juul, Ala Trusina

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.

The article can be found here.

## Rational and actual behavior in lowest unique bid auctions

By Simone Pigolotti, Sebastian Bernhardsson, Jeppe Juul, Gorm Galster, and Pierpaolo Vivo

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 derive an analytical expression for the equilibrium distribution of the game as a function of N and study its properties, which are then compared with a large dataset of internet auctions. The empirical collective strategy reproduces the theoretical equilibrium with striking accuracy for small N, while for larger N the quality of the fit deteriorates. As a consequence, the same game exhibits lottery-like and game-of-skill features, depending on the collective size of the bidding pool. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.

The article can be found here.

## Locally self-organized quasi-critical percolation in a multiple disease model

By Jeppe Juul and Kim Sneppen

Diseases emerge, persist and vanish in an ongoing battle for available hosts. Hosts, on the other hand, defend themselves by developing immunity that limits the ability of pathogens to reinfect them. We here explore a multi-disease system with emphasis on mutual exclusion. We demonstrate that such a system develops towards a steady state, where the spread of individual diseases self-organizes to a state close to that of critical percolation, without any global control mechanism or separation of time scale. For a broad range of introduction rates of new diseases, the likelihood of transmitting diseases remains approximately constant.

The article can be found here.

## AC-losses in Bi2Sr2Ca2Cu3O10+x tapes and a 3.15 meters long single phase cable

By Jeppe Juul, Esben Mølgaard, Jens Jensen, Niels Hessel Andersen, Asger Bech Abrahamsen, Dag Willen, Chresten Træholt, Carsten Thidemann, Heidi Lentge

The AC losses in superconducting multifilament BiSCCO-2223 tapes and a 3.15 m single phase test cable were measured at 77 K using an electrical transport method. The cable had an inner diameter of 42 mm, it was composed of a single layer of 31 multifilament tapes and had a critical current of Ic=4.1 kA. The measured losses of the tapes were found to be in good agreement with the Norris ellipse model. The losses of the cable were, for high currents, found to be bounded by the monoblock and independent Norris ellipse models.

You can find the article online here or download it here