Fixation times in graph-structured populations

The Moran process is widely used for modeling stochastic dynamics of finitely large populations. It describes the invasion process of a novel mutant into a resident population. Generally, the population is assumed to be well-mixed, which is a rather strong assumption. Studying the Moran process on graphs instead of unstructured populations is a recent approach to overcome this assumption. Some graph structures increase the fixation probability of a mutant that has a fitness advantage compared to the resident population. Graphs with this property are called amplifiers of selection. However, simulations show that the time until fixation increases considerably on those graphs. The objective of this thesis is to analyze different graphs of small size with respect to the fixation time. Simulations support the results for larger population size, where analytical approaches are unfeasible. We show that depending on the initial graph structure, the removal of one link can either lead to an increase or decrease in fixation time. This result is surprising and counterintuitive. Another interesting finding is that the shortest average fixation time does not only depend on the mutant’s starting node. But instead, different starting nodes are preferable, depending on the mutant’s fitness.1 Introduction 1 2 Background and Methods 5 2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The Moran Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 Fixation Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Isothermal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.3 Amplification and Suppression of Selection . . . . . . . . . . . . . . . . . . 13 2.3.4 Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.5 Effective Rate of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Small Population Size 17 3.1 Graph Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Fixation Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.2 Analytical and Simulated Fixation Probability . . . . . . . . . . . . . . . 27 3.3 Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Analytical Results for Fixation Time . . . . . . . . . . . . . . . . . . . . . 31 3.3.2 Simulation of Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Sojourn Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Effective Rate of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Location of the First Mutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Larger Graphs 45 4.1 Size Eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Removal of One and Two Links . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1.2 Removal of Three Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Influence of the "Five Links" on Fixation Time . . . . . . . . . . . . . . . . . . . 48 5 Discussion 49 5.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References 5

Should tissue structure suppress or amplify selection to minimize cancer risk?

It has been frequently argued that tissues evolved to suppress the accumulation of growth enhancing cancer inducing mutations. A prominent example is the hierarchical structure of tissues with high cell turnover, where a small number of tissue specific stem cells produces a large number of specialized progeny during multiple differentiation steps. Another well known mechanism is the spatial organization of stem cell populations and it is thought that this organization suppresses fitness enhancing mutations. However, in small populations the suppression of advantageous mutations typically also implies an increased accumulation of deleterious mutations. Thus, it becomes an important question whether the suppression of potentially few advantageous mutations outweighs the combined effects of many deleterious mutations

Wnt/β-catenin signalling induces MLL to create epigenetic changes in salivary gland tumours

We show that activation of Wnt/{beta}-catenin and attenuation of Bmp signals, by combined gain- and loss-of-function mutations of {beta}-catenin and Bmpr1a, respectively, results in rapidly growing, aggressive squamous cell carcinomas (SCC) in the salivary glands of mice. Tumours contain transplantable and hyperproliferative tumour propagating cells, which can be enriched by fluorescence activated cell sorting (FACS). Single mutations stimulate stem cells, but tumours are not formed. We show that {beta}-catenin, CBP and Mll promote self-renewal and H3K4 tri-methylation in tumour propagating cells. Blocking {beta}-catenin-CBP interaction with the small molecule ICG-001 and small-interfering RNAs against {beta}-catenin, CBP or Mll abrogate hyperproliferation and H3K4 tri-methylation, and induce differentiation of cultured tumour propagating cells into acini-like structures. ICG-001 decreases H3K4me3 at promoters of stem cell-associated genes in vitro and reduces tumour growth in vivo. Remarkably, high Wnt/{beta}-catenin and low Bmp signalling also characterize human salivary gland SCC and head and neck SCC in general. Our work defines mechanisms by which {beta}-catenin signals remodel chromatin and control induction and maintenance of tumour propagating cells. Further, it supports new strategies for the therapy of solid tumours

Fast flowing populations are not well mixed

In evolutionary dynamics, well-mixed populations are almost always associated with all-to-all interactions; mathematical models are based on complete graphs. In most cases, these models do not predict fixation probabilities in groups of individuals mixed by flows. We propose an analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.Comment: 19 pages, 8 figure

Scale-up and large-scale production of Tetraselmis sp CTP4 (Chlorophyta) for CO2 mitigation: from an agar plate to 100-m(3) industrial photobioreactors

Industrial production of novel microalgal isolates is key to improving the current portfolio of available strains that are able to grow in large-scale production systems for different biotechnological applications, including carbon mitigation. In this context, Tetraselmis sp. CTP4 was successfully scaled up from an agar plate to 35-and 100-m(3) industrial scale tubular photobioreactors (PBR). Growth was performed semi-continuously for 60 days in the autumn-winter season (17th October -14th December). Optimisation of tubular PBR operations showed that improved productivities were obtained at a culture velocity of 0.65-1.35 m s(-1) and a pH set-point for CO2 injection of 8.0. Highest volumetric (0.08 +/- 0.01 g L-1 d(-1)) and areal (20.3 +/- 3.2 g m(-2) d(-1)) biomass productivities were attained in the 100-m(3) PBR compared to those of the 35-m(3) PBR (0.05 +/- 0.02 g L-1 d(-1) and 13.5 +/- 4.3 g m(-2) d(-1), respectively). Lipid contents were similar in both PBRs (9-10% of ash free dry weight). CO2 sequestration was followed in the 100-m(3) PBR, revealing a mean CO2 mitigation efficiency of 65% and a biomass to carbon ratio of 1.80. Tetraselmis sp. CTP4 is thus a robust candidate for industrial-scale production with promising biomass productivities and photosynthetic efficiencies up to 3.5% of total solar irradiance.Portuguese national budget; Foundation for Science and Technology (FCT) [CCMAR/Multi/04326/2013]; INTERREG V-A Espana-Portugal project [0055 ALGARED + 5 E]; COST Action - European Network for Bio-products [1408]; FCT [SFRH/BD/105541/2014]; Nord Universityinfo:eu-repo/semantics/publishedVersio

Markers of thrombogenesis are activated in unmedicated patients with acute psychosis: a matched case control study

<p>Abstract</p> <p>Background</p> <p>Antipsychotic treatment has been repeatedly found to be associated with an increased risk for venous thromboembolism in schizophrenia. The extent to which the propensity for venous thromboembolism is linked to antipsychotic medication alone or psychosis itself is unclear. The objective of this study was to determine whether markers of thrombogenesis are increased in psychotic patients who have not yet been treated with antipsychotic medication.</p> <p>Methods</p> <p>We investigated the plasma levels of markers indicating activation of coagulation (D-dimers and Factor VIII) and platelets (soluble P-selectin, sP-selectin) in an antipsychotic-naive group of fourteen men and eleven women with acute psychosis (age 29.1 ± 8.3 years, body mass index 23.6 ± 4.7), and twenty-five healthy volunteers were matched for age, gender and body mass index.</p> <p>Results</p> <p>D-dimers (median 0.38 versus 0.19 mg/l, mean 1.12 ± 2.38 versus 0.28 ± 0.3 mg/l; P = 0.003) and sP-selectin (median 204.1 versus 112.4 ng/ml, mean 209.9 ± 124 versus 124.1 ± 32; P = 0.0005) plasma levels were significantly increased in the group of patients with acute psychosis as compared with healthy volunteers. We found a trend (median 148% versus 110%, mean 160 ± 72.5 versus 123 ± 62.5; P = 0.062) of increased plasma levels of factor VIII in psychotic patients as compared with healthy volunteers.</p> <p>Conclusions</p> <p>The results suggest that at least a part of venous thromboembolic events in patients with acute psychosis may be induced by pathogenic mechanisms related to psychosis rather than by antipsychotic treatment. Finding an exact cause for venous thromboembolism in psychotic patients is necessary for its effective treatment and prevention.</p

Reward and punishment in climate change dilemmas

Mitigating climate change effects involves strategic decisions by individuals that may choose to limit their emissions at a cost. Everyone shares the ensuing benefits and thereby individuals can free ride on the effort of others, which may lead to the tragedy of the commons. For this reason, climate action can be conveniently formulated in terms of Public Goods Dilemmas often assuming that a minimum collective effort is required to ensure any benefit, and that decision-making may be contingent on the risk associated with future losses. Here we investigate the impact of reward and punishment in this type of collective endeavors - coined as collective-risk dilemmas - by means of a dynamic, evolutionary approach. We show that rewards (positive incentives) are essential to initiate cooperation, mostly when the perception of risk is low. On the other hand, we find that sanctions (negative incentives) are instrumental to maintain cooperation. Altogether, our results are gratifying, given the a-priori limitations of effectively implementing sanctions in international agreements. Finally, we show that whenever collective action is most challenging to succeed, the best results are obtained when both rewards and sanctions are synergistically combined into a single policy.This research was supported by Fundacao para a Ciencia e Tecnologia (FCT) through grants PTDC/EEISII/5081/2014 and PTDC/MAT/STA/3358/2014 and by multiannual funding of INESC-ID and CBMA (under the projects UID/CEC/50021/2019 and UID/BIA/04050/2013). F.P.S. acknowledges support from the James S. McDonnell Foundation 21st Century Science Initiative in Understanding Dynamic and Multi-scale Systems Postdoctoral Fellowship Award. All authors declare no competing financial or non-financial interests in relation to the work described

The effect of graph structure on the dynamics of a stochastic evolutionary process

Evolutionary graph theory is the study of how spatial population structure affects evolutionary processes. The nodes of the graph are inhabited by individuals, e.g. cells. The links between nodes represent possibilities for these individuals to spread. We study the Moran process, where there is one birth and death event per time step. In the most commonly used updating mechanism, birth happens with probability proportional to the individual’s fitness. The individual giving birth then randomly replaces one of its neighbors by an identical copy of itself. Initially, the network is inhabited entirely by wild-type individuals with fitness 1 and one mutant with relative fitness r > 0. One interesting property of this system is the probability with which the mutant will give rise to a lineage that takes over the whole network, the so-called fixation probability. There are certain networks that can increase or decrease this probability compared to the unstructured case, called amplifiers and suppressors of selection, respectively. We find that most small undirected random graphs are amplifiers of selection. If we however change the updating rule to remove a random individual first and subsequently let its neighbors compete for the empty slot according to their fitness, this completely changes the result. Under death-birth updating, almost all undirected random graphs are suppressors of selection. Another evolutionary outcome of interest is the expected time this process takes until the mutants fixate in the population. Since it is known that certain amplifier graphs also increase the time to fixation, we are interested in the specific effect of graph structure on fixation time. We show that this fixation time can both increase or decrease when removing a link from a graph. Often, the fixation probability and time are either calculated analytically for simple cases or simulated for larger or more complicated graphs. We use standard Markov chain methods to numerically solve the system which has advantages over both analytical calculations and simulations. For this, we provide code to automate the part of creating the transition matrix for arbitrary graph structure. Lastly, we apply this abstract model to a conceptual question in biology, namely to cancer initiation. We are interested to find a graph which would provide an optimal tissue structure to prevent cancer mutations from spreading through the whole graph. Surprisingly, we conclude that it is not always the strongest suppressor of selection that works best at preventing this. But instead it highly depends on the fitness distribution of newly arising mutations and on the detailed update mechanism.Contents 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Evolutionary graph theory . . . . . . . . . . . . . . . . . . . . 3 1.2.1 The Moran process . . . . . . . . . . . . . . . . . . . . 3 1.2.2 The Moran process on graphs . . . . . . . . . . . . . . 5 1.2.3 Amplification and suppression of selection . . . . . . . 6 1.3 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . 9 2 Counterintuitive properties of the fixation time 11 2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 The Moran process in well-mixed populations . . . . . 13 2.2.2 The Moran process in structured populations . . . . . 14 2.2.3 A general approach to calculate probabilities and times of fixation . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Analytical results for small networks . . . . . . . . . . 17 2.3.2 Numerical simulations for larger networks . . . . . . . 28 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Amplifiers and suppressors of selection 33 3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Fixation probabilities in well-mixed populations . . . . 36 3.3.2 Numerical procedure . . . . . . . . . . . . . . . . . . . 38 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.1 Birth-death . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.2 death-Birth . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.3 Directed graphs . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Numerical method and algorithm 51 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Software description . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.1 Computing the transition matrix from the adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.2 Fixation probability . . . . . . . . . . . . . . . . . . . 55 4.3.3 Fixation time . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.4 Computational limitations and performance . . . . . . 58 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Application to a question in cancer initiation 63 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3 Fixation of novel mutations . . . . . . . . . . . . . . . . . . . 66 5.4 The distribution of fitness effects of cancer mutations . . . . . 69 5.5 Population structures and their effect on fixation probabilities 72 5.6 Double mutations . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6 Discussion 81 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Bibliography 9