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• 1.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
A combined discontinuous Galerkin and finite volume scheme for multi–dimensional VPFP system2011In: AIP Conference Proceedings 1333 / [ed] Deborah A. Levin, Ingrid J. Wysong, Alejandro L. Garcia and Henry Abarbanel, American Institute of Physics (AIP), 2011, Vol. 1333, p. 57-62Conference paper (Refereed)

We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.

• 2.
Gdansk University of Technology, Poland.
A Graph – String Model of Gene Assembly in Ciliates [Grafowo-tekstowy model rekombinacji DNA u orzęsek]2006In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2006, p. 521-534Conference paper (Refereed)

The ciliates are a family of unicellular organisms that characterize themselves by having two types of nuclei, micro - and macronuclei. During cell mating the genetic material must change from the micronuclei to the macronuclei form. The paper summarises a formal model for this change. The model, which is described in recent works, is based on strings and graphs. It shows that inside the cell complex computational operations have to take place.

• 3.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Exact and approximate limit behaviour of the Yule trees cophenetic index2018In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 303, p. 26-45Article in journal (Refereed)

In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure-birth tree conditioned on n contemporary tips. We show that this normalized phylogenetic balance index is a sub-martingale that converges almost surely and in L-2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction-type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure-birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

• 4.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Limit distribution of the quartet balance index for Aldous’s $(\beta \ge 0)$-model2019In: Applicationes Mathematicae, ISSN 1233-7234, E-ISSN 1730-6280Article in journal (Refereed)

This paper builds on T. Martínez-Coronado, A. Mir, F. Rosselló and G. Valiente’s 2018 work, introducing a new balance index for trees. We show that this balance index, in the case of Aldous’s $(\beta \ge 0)$-model, converges weakly to a distribution that can be characterized as the fixed point of a contraction operator on a class of distributions.

• 5.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Phylogenetic effective sample size2016In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 407, p. 371-386Article in journal (Refereed)

In this paper I address the question—how large is a phylogenetic sample? I propose a definition of a phylogenetic effective sample size for Brownian motion and Ornstein-Uhlenbeck processes-the regression effective sample size. I discuss how mutual information can be used to define an effective sample size in the non-normal process case and compare these two definitions to an already present concept of effective sample size (the mean effective sample size). Through a simulation study I find that the AICc is robust if one corrects for the number of species or effective number of species. Lastly I discuss how the concept of the phylogenetic effective sample size can be useful for biodiversity quantification, identification of interesting clades and deciding on the importance of phylogenetic correlations.

• 6.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Quantifying the effects of anagenetic and cladogenetic evolution2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 254, p. 42-57Article in journal (Refereed)

An ongoing debate in evolutionary biology is whether phenotypic change occurs predominantly around the time of speciation or whether it instead accumulates gradually over time. In this work I propose a general framework incorporating both types of change, quantify the effects of speciational change via the correlation between species and attribute the proportion of change to each type. I discuss results of parameter estimation of Hominoid body size in this light. I derive mathematical formulae related to this problem, the probability generating functions of the number of speciation events along a randomly drawn lineage and from the most recent common ancestor of two randomly chosen tip species for a conditioned Yule tree. Additionally I obtain in closed form the variance of the distance from the root to the most recent common ancestor of two randomly chosen tip species.

• 7.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Simulating an infinite mean waiting time2019In: Mathematica Applicanda, ISSN 1730-2668, Vol. 47, no 1, p. 93-102Article in journal (Refereed)

We consider a hybrid method to simulate the return time to the initial state in a critical-case birth-death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.

• 8.
Gdansk University of Technology, Poland.
The Bootstrap and Other Methods of Testing Phylogenetic Trees2007In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2007, p. 103-108Conference paper (Refereed)

The final step of a phylogenetic analysis is the test of the generated tree. This is not a easy task for which there is an obvious methodology because we do not know the full probabilistic model of evolution. A number of methods have been proposed but there is a wide debate concerning the interpretations of the results they produce.

• 9.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
The phylogenetic effective sample size and jumps2018In: MATHEMATICA APPLICANDA (MATEMATYKA STOSOWANA), ISSN 1730-2668, Vol. 46, no 1, p. 25-33Article in journal (Refereed)

The phylogenetic effective sample size is a parameter that has as its goal the quantification of the amount of independent signal in a phylogenetically correlatedsample. It was studied for Brownian motion and Ornstein-Uhlenbeck models of trait evolution. Here, we study this composite parameter when the trait is allowedto jump at speciation points of the phylogeny. Our numerical study indicates thatthere is a non-trivial limit as the effect of jumps grows. The limit depends on thevalue of the drift parameter of the Ornstein-Uhlenbeck process.

• 10.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Trait evolution with jumps: illusionary normality2017In: Proceedings of the XXIII National Conference on Applications of Mathematics in Biology and Medicine, 2017, p. 23-28Conference paper (Refereed)

Phylogenetic comparative methods for real-valued traits usually make use of stochastic process whose trajectories are continuous.This is despite biological intuition that evolution is rather punctuated thangradual. On the other hand, there has been a number of recent proposals of evolutionarymodels with jump components. However, as we are only beginning to understandthe behaviour of branching Ornstein-Uhlenbeck (OU) processes the asymptoticsof branching  OU processes with jumps is an even greater unknown. In thiswork we build up on a previous study concerning OU with jumps evolution on a pure birth tree.We introduce an extinction component and explore via simulations, its effects on the weak convergence of such a process.We furthermore, also use this work to illustrate the simulation and graphic generation possibilitiesof the mvSLOUCH package.

• 11.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
A Noether theorem for stochastic operators on Schatten classes2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 2, p. 1395-1412Article in journal (Refereed)

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

• 12.
Gdansk University of Technology, Poland.
Gdansk University of Technology, Poland.
On the Time Behaviour of Okazaki Fragments2006In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 43, no 2, p. 500-509Article in journal (Refereed)

We find explicit analytical formulae for the time dependence of the probability of the number of Okazaki fragments produced during the process of DNA replication. This extends a result of Cowan on the asymptotic probability distribution of these fragments.

• 13.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
State Univ Appl Sci Elblag, Krzysztof Brzeski Inst Appl Informat, Ul Wojska Polskiego 1, PL-82300 Elblag, Poland. Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland.
Weak Stability of Centred Quadratic Stochastic Operators2019In: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, Vol. 42, no 4, p. 1813-1830Article in journal (Refereed)

We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offsprings trait is equal to an additively perturbed arithmetic mean of the parents traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

• 14.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala University, Sweden.
Uppsala University, Sweden; CNRS University of Montpellier IRD EPHE, France. Uppsala University, Sweden. Uppsala University, Sweden.
Using the Ornstein-Uhlenbeck process to model the evolution of interacting populations2017In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 429, p. 35-45Article in journal (Refereed)

The Ornstein-Uhlenbeck (OU) process plays a major role in the analysis of the evolution of phenotypic traits along phylogenies. The standard OU process includes random perturbations and stabilizing selection and assumes that species evolve independently. However, evolving species may interact through various ecological process and also exchange genes especially in plants. This is particularly true if we want to study phenotypic evolution among diverging populations within species. In this work we present a straightforward statistical approach with analytical solutions that allows for the inclusion of adaptation and migration in a common phylogenetic framework, which can also be useful for studying local adaptation among populations within the same species. We furthermore present a detailed simulation study that clearly indicates the adverse effects of ignoring migration. Similarity between species due to migration could be misinterpreted as very strong convergent evolution without proper correction for these additional dependencies. Finally, we show that our model can be interpreted in terms of ecological interactions between species, providing a general framework for the evolution of traits between "interacting" species or populations.(C) 2017 Elsevier Ltd. All rights reserved.

• 15.
Gdansk University of Technology.
Gdansk University of Technology. Gdansk University of Technology, Poland. Medical University of Gdansk, Poland. Gdansk University of Technology, Poland. Gdansk University of Technology, Poland.
Neural Network Breast Cancer Relapse Time Prognosis2006In: ASO Summer School 2006 abstract book Ostrzyce 30.06-2.07. 2006 / [ed] J. Skokowski and K. Drucis, 2006, p. 8-10Conference paper (Other academic)

This paper is a result of a project at the Faculty of Electronics, Telecommunication and Computer Science (Technical University of Gdansk). The aim of the project was to create a neural network to predict the relapsetime of breast cancer. The neural network was to be trained on data collected over the past 20 years by dr. Jarosław Skokowski. The data includes 439 patient records described by about 40 parameters. For our neuralnetwork we only considered 6 medically most significant parameters the number of nodes showing evidence of cancer, size of tumour (in mm.), age, bloom score, estrogen receptors and proestrogen receptors and the relapsetime as the outcome. Our neural network was created in the MATLAB environment.

• 16.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden / Department of Biological and Environmental Science, University of Gothenburg, Gothenburg, Sweden. Department of Biological and Environmental Science, University of Gothenburg, Gothenburg, Sweden. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Time to a single hybridization event in a group of species with unknown ancestral history2013In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 322, p. 1-6Article in journal (Refereed)

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments, and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

• 17.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden.
Gdansk University of Technology.
Critical case stochastic phylogenetic tree model via the Laplace transform2014In: Demonstratio Matematicae, ISSN 0420-1213, Vol. 47, no 2, p. 474-481Article in journal (Refereed)

Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both in uenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this we correct the proof of Liggett and Schinazi (2009) in the critical case.

• 18.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Gdansk University of Technology. Medical University of Gdansk.
Survival time prognosis under a Markov model of cancer development2010In: Proceedings of the XVI National Conference Applications of Mathematics to Biology and Medicine, Krynica, Poland, September 14–18, 2010 / [ed] M. Ziółko, M. Bodnar and E. Kutafina, 2010, p. 6-11Conference paper (Refereed)

In this study we look at a breast cancer data set of women from the Pomerania region collected in the year 1987- 1992 in the Medical University of Gdansk.We analyze the clinical risk factors in conjunction with a Markov model of cancer development. We evaluate Artificial Neural Network (ANN) survival time prediction (which was done on this data set in a previous study) via a simulation study.

• 19.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Univ Cambridge, England.
MODELLING TRAIT-DEPENDENT SPECIATION WITH APPROXIMATE BAYESIAN COMPUTATION2019In: ACTA PHYSICA POLONICA B PROCEEDINGS SUPPLEMENT, JAGIELLONIAN UNIV , 2019, Vol. 12, no 1, p. 25-47Conference paper (Refereed)

Phylogeny is the field of modelling the temporal discrete dynamics of speciation. Complex models can nowadays be studied using the Approximate Bayesian Computation approach which avoids likelihood calculations. The fields progression is hampered by the lack of robust software to estimate the numerous parameters of the speciation process. In this work, we present an R package, pcmabc, publicly available on CRAN, based on Approximate Bayesian Computations, that implements three novel phylogenetic algorithms for trait-dependent speciation modelling. Our phylogenetic comparative methodology takes into account both the simulated traits and phylogeny, attempting to estimate the parameters of the processes generating the phenotype and the trait. The user is not restricted to a predefined set of models and can specify a variety of evolutionary and branching models. We illustrate the software with a simulation-reestimation study focused around the branching Ornstein-Uhlenbeck process, where the branching rate depends non-linearly on the value of the driving Ornstein-Uhlenbeck process. Included in this work is a tutorial on how to use the software.

• 20.
Mathematical Statistics, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
Computer Laboratory, University of Cambridge Cambridge, United Kingdom. Computer Laboratory, University of Cambridge Cambridge, United Kingdom.
Influenza differentiation and evolution2010In: Acta Physica Polonica B Proceedings Supplement, 2010, Vol. 3, p. 417-452, article id 2Conference paper (Refereed)

The aim of the study is to do a very wide analysis of HA, NA and M influenza gene segments to find short nucleotide regions,which differentiate between strains (i.e. H1, H2, ... e.t.c.), hosts, geographic regions, time when sequence was found and combination of time and region using a simple methodology. Finding regions  differentiating between strains has as its goal the construction of a Luminex microarray which will allow quick and efficient strain recognition. Discovery for the other splitting factors could shed lighton structures significant for host specificity and on the history of influenza evolution. A large number of places in the HA, NA and M gene segments were found that can differentiate between hosts, regions, time and combination of time and region. Also very good differentiation between different Hx strains can be seen.We link one of our findings to a proposed stochastic model of creation of viral phylogenetic trees.

• 21.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala Univ, Sweden.
Polish Acad Sci, Poland. Univ Lodz, Poland. Polish Acad Sci, Poland. Uppsala Univ, Sweden. Polish Acad Sci, Poland.
Predicting pathogenicity behavior in Escherichia coli population through a state dependent model and TRS profiling2018In: PloS Computational Biology, ISSN 1553-734X, E-ISSN 1553-7358, Vol. 14, no 1, article id e1005931Article in journal (Refereed)

The Binary State Speciation and Extinction (BiSSE) model is a branching process based model that allows the diversification rates to be controlled by a binary trait. We develop a general approach, based on the BiSSE model, for predicting pathogenicity in bacterial populations from microsatellites profiling data. A comprehensive approach for predicting pathogenicity in E. coli populations is proposed using the state-dependent branching process model combined with microsatellites TRS-PCR profiling. Additionally, we have evaluated the possibility of using the BiSSE model for estimating parameters from genetic data. We analyzed a real dataset (from 251 E. coli strains) and confirmed previous biological observations demonstrating a prevalence of some virulence traits in specific bacterial sub-groups. The method may be used to predict pathogenicity of other bacterial taxa.

• 22.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Department of Genetics, University of Pretoria, Pretoria 0002, South Africa. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden. Department of Biological and Environmental Sciences, University of Gothenburg, Gothenburg, Sweden. CEES, Department of Biology, University of Oslo, Oslo, Norway.
A phylogenetic comparative method for studying multivariate adaptation2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 314, p. 204-215Article in journal (Refereed)

Phylogenetic comparative methods have been limited in the way they model adaptation. Although some progress has been made, there are still no methods that can fully account for coadaptationbetween traits. Based on Ornstein-Uhlenbeck (OU) models of adaptive evolution, we present a method,with R implementation, in which multiple traits evolve both in response to each other and, as inprevious OU models, to fixed or randomly evolving predictor variables. We present the interpretation ofthe model parameters in terms of evolutionary and optimal regressions enabling the study of allometric and adaptive relationships between traits. To illustrate the method we reanalyze a data set of antlerand body-size evolution in deer (Cervidae).

• 23.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Computer Laboratory , University of Cambridge, Cambridge, Un ited Kingdom.
A novel algorithm to reconstruct phylogenies using gene sequences and expression data2014In: International Proceedings of Chemical, Biological & Environmental Engineering; Environment, Energy and Biotechnology III, 2014, Vol. 70, p. 8-12Conference paper (Refereed)

Phylogenies based on single loci should be viewed with caution and the best approach for obtaining robust trees is to examine numerous loci across the genome. It often happens that for the same set of species trees derived from different genes are in conflict between each other. There are several methods that combine information from different genes in order to infer the species tree. One novel approach is to use informationfrom different -omics. Here we describe a phylogenetic method based on an Ornstein–Uhlenbeck process that combines sequence and gene expression data. We test our method on genes belonging to the histidine biosynthetic operon. We found that the method provides interesting insights into selection pressures and adaptive hypotheses concerning gene expression levels.

• 24.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden.
Department of Probability and Biomathematics, Gdánsk University of Technology, Gdánsk, Poland.
Quadratic stochastic operators as a tool in modelling the dynamics of a distribution of a population trait2013In: Proceedings of the 19th National Conference on Applications of Mathematics in Biology and Medicine / [ed] Katarzyna D. Lewandowska and Piotr Bogús, 2013, p. 19-24Conference paper (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours and these have been introducedand studied recently. In the present work we discuss biological interpretations that can be attributedto them. We also propose a computer simulation method to illustrate the behaviour of iterates of quadratic stochastic operators.

• 25.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
Asymptotic properties of quadratic stochastic operators acting on the L1 space2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 114, p. 26-39Article in journal (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space. It turns out that inprinciple most of the results can be carried over to the L1 space. However, due to topologicalproperties of this space one has to restrict in some situations to kernel quadratic stochasticoperators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1 space in terms of convergence of the associated (linear)nonhomogeneous Markov chains.

• 26.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
Prevalence Problem in the Set of Quadratic StochasticOperators Acting on L12018In: Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, Vol. 41, no 1, p. 159-173Article in journal (Refereed)

This paper is devoted to the study of the problem of prevalence in the classof quadratic stochastic operators acting on the L1 space for the uniform topology.We obtain that the set of norm quasi-mixing quadratic stochastic operators is a denseand open set in the topology induced by a very natural metric. This shows the typicallong-term behaviour of iterates of quadratic stochastic operators.

• 27.
Uppsala universitet, Tillämpad matematik och statistik.
A consistent estimator of the evolutionary rate2015In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 371, p. 69-78Article in journal (Refereed)

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate 2, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance sigma(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator 4 of the evolutionary rate rho(2) - sigma(2)/lambda. The estimator R-n(2) is proportional to the sample variance S-n(2) computed from n trait values. We find an approximate formula for the standard error of R-n(2), based on a neat asymptotic relation for the variance of S-n(2). (C) 2015 Elsevier Ltd. All rights reserved.

• 28.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Chalmers University of Technology and the Unversity of Gothenburg, Sweden.
Phylogenetic confidence intervals for the optimal trait value2015In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 52, no 4, p. 1115-1132Article in journal (Refereed)

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree ismodelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein–Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the samplemean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean isapproximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

• 29.
Gdansk University of Technology, Poland.
Gdansk University of Technology, Poland.
Moments of the Distribution of Okazaki Fragments2006In: Rose–Hulman Undergraduate Mathematics Journal, Vol. 7, no 2, p. 1-5Article in journal (Refereed)

This paper is a continuation of Bartoszek & Bartoszek (2006) who provide formulae for the probability distributions of the number of Okazaki fragments at time t during the process of DNA replication. Given the expressions for the moments of the probability distribution of the number of Okazaki fragments at time t in the recursive form, we evaluated formulae for the third and fourth moments, using Mathematica, and obtained results in explicit form. Having done this, we calculated the distribution’s skewness and kurtosis.

• 30.
Department of Mathematics, Gdansk University of Technology, Gdansk, Poland.
Department of Mathematics, Gdansk University of Technology, Gdansk, Poland.
The Fundamental Group, Covering Spaces and Topology in Biology2006In: Conference materials 9th International Workshop for Young Mathematicians "Topology", 2006, p. 19-29Conference paper (Other academic)

Abstract. We give a short introduction to homotopy theory. We pass to the concepts of a pointed space (X, x0), the fundamental group of X, a simply connected space (with the example of the space contractible to a point), introduce basic concepts of covering spaces (e.g. covering map/space, fiber over x, Path lifting Theorem). With the use of the exponential map and the idea of the index of a loop, we show that the fundamental group of the circle S1 is isomorphic to the integers Z with addition. We mention some other interesting fundamental groups (e.g. the fundamental group of a torus or of thefigure eight). We also present some very interesting applications of topological concepts in Molecular Biology.

• 31.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
University of Gothenburg.
Performance of pseudo-likelihood estimator in modelling cells' proliferation with noisy measurements2010In: Conference proceedings from the 12th International Workshop for Young Mathematicians "Probability and Statistics", Krakow, Poland, 20th till 26th September 2009, 2010, p. 21-42Conference paper (Other academic)

Branching processes are widely used to describe cell development and proliferation. Currently parameter estimation is studied in mathematical models describing the dynamics of cell cultures where we can get very accurate measurements of cell counts. In vivo samples we will not have this accuracy, here the noise levels can be very significant. We will study a newly proposed pseudo-likelihood estimator of a multitype Bellman-Harris process modelling cell development and see how it performs under noisy measurements of cell counts.

• 32.
Harvard Univ, MA 02138 USA.
Harvard Univ, MA 02138 USA; Gothenburg Global Biodivers Ctr, Sweden; Univ Gothenburg, Sweden; Gothenburg Bot Garden, Sweden. Gothenburg Global Biodivers Ctr, Sweden; Univ Gothenburg, Sweden. Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Swedish Museum Nat Hist, Sweden. Univ Neuchatel, Switzerland. Univ Gothenburg, Sweden. Univ Michigan, MI 48109 USA. Harvard Univ, MA 02138 USA. Univ Oslo, Norway. Ecole Normale Super Paris, France. Rice Univ, TX USA. Gothenburg Global Biodivers Ctr, Sweden; Univ Gothenburg, Sweden. Univ Gothenburg, Sweden. Chalmers Univ Technol, Sweden; Univ Gothenburg, Sweden. Lund Univ, Sweden. Inst Nacl de Pesquisas da Amazonia, Brazil. Chalmers Univ Technol, Sweden; Univ Gothenburg, Sweden; Rutgers State Univ, NJ USA. Univ Kwazulu Natal, South Africa. Harvard Univ, MA 02138 USA; Univ Gothenburg, Sweden; Chalmers Univ Technol, Sweden.
Embracing heterogeneity: coalescing the Tree of Life and the future of phylogenomics2019In: PeerJ, ISSN 2167-8359, E-ISSN 2167-8359, Vol. 7, article id e6399Article in journal (Refereed)

Building the Tree of Life (ToL) is a major challenge of modern biology, requiring advances in cyberinfrastructure, data collection, theory, and more. Here, we argue that phylogenomics stands to benefit by embracing the many heterogeneous genomic signals emerging from the first decade of large-scale phylogenetic analysis spawned by high-throughput sequencing (HTS). Such signals include those most commonly encountered in phylogenomic datasets, such as incomplete lineage sorting, but also those reticulate processes emerging with greater frequency, such as recombination and introgression. Here we focus specifically on how phylogenetic methods can accommodate the heterogeneity incurred by such population genetic processes; we do not discuss phylogenetic methods that ignore such processes, such as concatenation or supermatrix approaches or supertrees. We suggest that methods of data acquisition and the types of markers used in phylogenomics will remain restricted until a posteriori methods of marker choice are made possible with routine whole-genome sequencing of taxa of interest. We discuss limitations and potential extensions of a model supporting innovation in phylogenomics today, the multispecies coalescent model (MSC). Macroevolutionary models that use phylogenies, such as character mapping, often ignore the heterogeneity on which building phylogenies increasingly rely and suggest that assimilating such heterogeneity is an important goal moving forward. Finally, we argue that an integrative cyberinfrastructure linking all steps of the process of building the ToL, from specimen acquisition in the field to publication and tracking of phylogenomic data, as well as a culture that values contributors at each step, are essential for progress.

• 33.
Department of Biology, Centre for Ecological and Evolutionary Synthesis, University of Oslo, Norway.
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Interpreting the evolutionary regression: The interplay between observational and biological errors in phylogenetic comparative studies2012In: Systematic Biology, ISSN 1063-5157, E-ISSN 1076-836X, Vol. 61, no 3, p. 413-425Article in journal (Refereed)

Regressions of biological variables across species are rarely perfect. Usually, there are residual deviations fromthe estimated model relationship, and such deviations commonly show a pattern of phylogenetic correlations indicatingthat they have biological causes. We discuss the origins and effects of phylogenetically correlated biological variation inregression studies. In particular, we discuss the interplay of biological deviations with deviations due to observationalor measurement errors, which are also important in comparative studies based on estimated species means. We showhow bias in estimated evolutionary regressions can arise from several sources, including phylogenetic inertia and eitherobservational or biological error in the predictor variables. We show how all these biases can be estimated and correctedfor in the presence of phylogenetic correlations.We present general formulas for incorporating measurement error in linearmodels with correlated data. We also show how alternative regression models, such as major axis and reduced major axisregression, which are often recommended when there is error in predictor variables, are strongly biased when there isbiological variation in any part of the model.We argue that such methods should never be used to estimate evolutionary orallometric regression slopes.

• 34.
Swiss Fed Inst Technol, Switzerland; Swiss Inst Bioinformat SIB, Switzerland.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Swiss Fed Inst Technol, Switzerland; Swiss Inst Bioinformat SIB, Switzerland.
Automatic generation of evolutionary hypotheses using mixed Gaussian phylogenetic models2019In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 116, no 34, p. 16921-16926Article in journal (Refereed)

Phylogenetic comparative methods are widely used to understand and quantify the evolution of phenotypic traits, based on phylogenetic trees and trait measurements of extant species. Such analyses depend crucially on the underlying model. Gaussian phylogenetic models like Brownian motion and Ornstein-Uhlenbeck processes are the workhorses of modeling continuous-trait evolution. However, these models fit poorly to big trees, because they neglect the heterogeneity of the evolutionary process in different lineages of the tree. Previous works have addressed this issue by introducing shifts in the evolutionary model occurring at inferred points in the tree. However, for computational reasons, in all current implementations, these shifts are "intramodel," meaning that they allow jumps in 1 or 2 model parameters, keeping all other parameters "global" for the entire tree. There is no biological reason to restrict a shift to a single model parameter or, even, to a single type of model. Mixed Gaussian phylogenetic models (MGPMs) incorporate the idea of jointly inferring different types of Gaussian models associated with different parts of the tree. Here, we propose an approximate maximum-likelihood method for fitting MGPMs to comparative data comprising possibly incomplete measurements for several traits from extant and extinct phylogenetically linked species. We applied the method to the largest published tree of mammal species with body-and brain-mass measurements, showing strong statistical support for an MGPM with 12 distinct evolutionary regimes. Based on this result, we state a hypothesis for the evolution of the brain-body-mass allometry over the past 160 million y.

• 35.
Chalmers University of Technology and the University of Gothenburg.
Uppsala universitet, Tillämpad matematik och statistik.
Convergence of Finite Volume Scheme for a Three-Dimensional Poisson Equation2014In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 202, no 2, p. 130-153Article in journal (Refereed)

We construct and analyze a finite volume scheme for numerical solution of a three-dimensional Poisson equation. We derive optimal convergence rates in the discrete H1 norm and sub-optimal convergence in the maximum norm, where we use the maximal available regularity of the exact solution and minimal smoothness requirement on the source term. The theoretical results are justified through implementing some canonical examples in 3D.

• 36.
Chalmers University of Technology and the University of Gothenburg.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Interspecies correlation for neutrally evolving traits2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 309, p. 11-19Article in journal (Refereed)

A simple way to model phenotypic evolution is to assume that after splitting, the trait values of the sister species diverge as independent Brownian motions. Relying only on a prior distribution for the underlying species tree (conditioned on the number, n, of extant species) we study the random vector (X1,…,Xn) of the observed trait values. In this paper we derive compact formulae for the variance of the sample mean and the mean of the sample variance for the vector (X1,…,Xn).

The key ingredient of these formulae is the correlation coefficient between two trait values randomly chosen from (X1,…,Xn). This interspecies correlation coefficient takes into account not only variation due to the random sampling of two species out of n and the stochastic nature of Brownian motion but also the uncertainty in the phylogenetic tree. The latter is modeled by a (supercritical or critical) conditioned branching process. In the critical case we modify the Aldous-Popovic model by assuming a proper prior for the time of origin.

• 37.
Department of Marine Botany, University of Bremen, Bremen, Germany / Section Functional Ecology, Department Seaweed Biology, Alfred-Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.
Section Ecological Chemistry, Alfred-Wegener Institute for Polar and Marine Research, Bremerhaven, Germany. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden. Department of Marine Botany, University of Bremen, Bremen, Germany. Section Functional Ecology, Department Seaweed Biology, Alfred-Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.
Phlorotannin Production and Lipid Oxidation as a Potential Protective Function Against High Photosynthetically Active and UV Radiation in Gametophytes of Alaria esculenta (Alariales, Phaeophyceae)2012In: Photochemistry and Photobiology, ISSN 0031-8655, E-ISSN 1751-1097, Vol. 88, no 1, p. 46-57Article in journal (Refereed)

Radiation damage can inter alia result in lipid peroxidation of macroalgal cell membranes. To prevent photo-oxidation within the cells, photoprotective substances such as phlorotannins are synthesized. In the present study, changes in total fatty acids (FA), FA composition and intra/extracellular phlorotannin contents were determined by gas chromatography and the Folin-Ciocalteu method to investigate the photoprotective potential of phlorotannins to prevent lipid peroxidation. Alaria esculenta juveniles (Phaeophyceae) were exposed over 20 days to high/low photosynthetically active radiation (PAR) in combination with UV radiation (UVR) in the treatments: PAB (low/high PAR + UV-B + UV-A), PA (low/high PAR + UV-A) or low/high PAR only. While extracellular phlorotannins increased after 10 days, intracellular phlorotannins increased with exposure time and PA and decreased under PAB. Interactive effects of time:radiation wavebands, time:PAR dose as well as radiation wavebands:PAR dose were observed. Low FA contents were detected in the PA and PAB treatments; interactive effects were observed between time:high PAR and PAB:high PAR. Total FA contents were correlated to extra/intracellular phlorotannin contents. Our results suggest that phlorotannins might play a role in intra/extracellular protection by absorption and oxidation processes. Changes in FA content/composition upon UVR and high PAR might be considered as an adaptive mechanism of the A. esculenta juveniles subjected to variations in solar irradiance.

• 38.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala University, Akademiska Sjukhuset, Uppsala, Sweden. Uppsala University, Akademiska Sjukhuset, Uppsala, Sweden.
PSICA: Decision trees for probabilistic subgroup identification with categorical treatments2019In: Statistics in Medicine, ISSN 0277-6715, E-ISSN 1097-0258, Vol. 38, no 22, p. 4436-4452Article in journal (Refereed)

Personalized medicine aims at identifying best treatments for a patient with given characteristics. It has been shown in the literature that these methods can lead to great improvements in medicine compared to traditional methods prescribing the same treatment to all patients. Subgroup identification is a branch of personalized medicine, which aims at finding subgroups of the patients with similar characteristics for which some of the investigated treatments have a better effect than the other treatments. A number of approaches based on decision trees have been proposed to identify such subgroups, but most of them focus on two‐arm trials (control/treatment) while a few methods consider quantitative treatments (defined by the dose). However, no subgroup identification method exists that can predict the best treatments in a scenario with a categorical set of treatments. We propose a novel method for subgroup identification in categorical treatment scenarios. This method outputs a decision tree showing the probabilities of a given treatment being the best for a given group of patients as well as labels showing the possible best treatments. The method is implemented in an R package psica available on CRAN. In addition to a simulation study, we present an analysis of a community‐based nutrition intervention trial that justifies the validity of our method.

• 39.
Department of Oncological Surgery, Medical University of Gdansk, Gdansk, Poland.
Department of Oncological Surgery, Medical University of Gdansk, Gdansk, Poland. 2Department of Mathematics, Uppsala University, Uppsala, Sweden. Department of Oncological Surgery, Medical University of Gdansk, Gdansk, Poland. Department of Medical Laboratory Diagnostics and Bank of Frozen Tissues and Genetic Specimens, Medical University of Gdansk, Gdansk, Poland. Department of Oncological Surgery, Medical University of Gdansk, Gdansk, Poland.
The impact of the Polish mass breast cancer screening program on prognosis in the Pomeranian Province2017In: Archives of Medical Science, ISSN 1734-1922, E-ISSN 1896-9151, Vol. 13, no 2, p. 441-447Article in journal (Refereed)

Introduction: Mammographic screening results in diagnosis of less advanced breast cancer (BC). A meta-analysis of randomized clinical trials confirmed that BC screening reduces mortality. In 2007, the National Breast Cancer Screening Program (NBCSP) was established in Poland with the crucial aim of reducing mortality from BC. The purpose of this study was to assess the impact of participation in the NBCSP on prognosis. Material and methods: A single institution, non-randomized retrospective study was undertaken. The study population comprised 643 patients with BC treated in the Department of Surgical Oncology (DSO) at the Medical University of Gdansk over a 4-year period, from 01.01.2007 until 31.12.2010. Patients were divided into two groups: group A-patients who participated in the NBCSP (n = 238, 37.0%); and group B-patients who did not participate in the NBCSP (n = 405, 63.0%). Results: Statistical analysis revealed that group A displayed a less advanced MCC stage (more patients in AJCC stage I, p = 0.002), lower tumor diameter (more patients with pT1, p = 0.006, and pT < 15 mm, p = 0.008) and a lower incidence of metastases to axillary lymph nodes (more patients with pNO, p = 0.01). From 2009 to 2010 the NBCSP revealed a statistically significant benefit significantly more patients in stage 0 + I (60.7% vs. 48.8%, p = 0.018) and with tumors pT < 15 mm (48.8% vs. 35.1%, p = 0.011) were observed in group A. Conclusions: The study results revealed the beneficial impact of the NBCSP. Superior prognostic factors and favorable staging were observed in women who participated in the NBCSP.

• 40.
Univ Cambridge, England.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Univ Cambridge, England.
Multi-omic analysis of signalling factors in inflammatory comorbidities2018In: BMC Bioinformatics, ISSN 1471-2105, E-ISSN 1471-2105, Vol. 19, article id 439Article in journal (Refereed)

BackgroundInflammation is a core element of many different, systemic and chronic diseases that usually involve an important autoimmune component. The clinical phase of inflammatory diseases is often the culmination of a long series of pathologic events that started years before. The systemic characteristics and related mechanisms could be investigated through the multi-omic comparative analysis of many inflammatory diseases. Therefore, it is important to use molecular data to study the genesis of the diseases. Here we propose a new methodology to study the relationships between inflammatory diseases and signalling molecules whose dysregulation at molecular levels could lead to systemic pathological events observed in inflammatory diseases.ResultsWe first perform an exploratory analysis of gene expression data of a number of diseases that involve a strong inflammatory component. The comparison of gene expression between disease and healthy samples reveals the importance of members of gene families coding for signalling factors. Next, we focus on interested signalling gene families and a subset of inflammation related diseases with multi-omic features including both gene expression and DNA methylation. We introduce a phylogenetic-based multi-omic method to study the relationships between multi-omic features of inflammation related diseases by integrating gene expression, DNA methylation through sequence based phylogeny of the signalling gene families. The models of adaptations between gene expression and DNA methylation can be inferred from pre-estimated evolutionary relationship of a gene family. Members of the gene family whose expression or methylation levels significantly deviate from the model are considered as the potential disease associated genes.ConclusionsApplying the methodology to four gene families (the chemokine receptor family, the TNF receptor family, the TGF- gene family, the IL-17 gene family) in nine inflammation related diseases, we identify disease associated genes which exhibit significant dysregulation in gene expression or DNA methylation in the inflammation related diseases, which provides clues for functional associations between the diseases.

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