Allele Frequency Spectrum in a Cancer Cell Population

Hisashi Ohtsuki, Hideki Innan

 

Abstract

A cancer grows from a single cell, thereby constituting a large cell population. In this work, we are interested in how mutations accumulate in a cancer cell population. We provide a theoretical framework of the stochastic process in a cancer cell population and obtain near exact expressions of allele frequency spectrum or AFS (only continuous approximation is involved) from both forward and backward treatments under a simple setting; all cells undergo cell division and die at constant rates, b and d, respectively, such that the entire population grows exponentially. This setting means that once a parental cancer cell is established, in the following growth phase, all mutations are assumed to have no effect on b or d (i.e., neutral or passengers). Our theoretical results show that the difference from organismal population genetics is mainly in the coalescent time scale, and the mutation rate is defined per cell division, not per time unit (e.g., generation). Except for these two factors, the basic logic are very similar between organismal and cancer population genetics, indicating that a number of well established theories of organismal population genetics could be translated to cancer population genetics with simple modifications.

doi: http://biorxiv.org/content/early/2017/01/30/104158 

 

 

Universal asymptotic clone size distribution for general population growth

Michael D. Nicholson, Tibor Antal

Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed as the Luria-Delbruck or Lea-Coulson model, is often assumed but seldom realistic. In this article we generalise the model to different types of wild-type population growth, with mutants evolving as a birth-death branching process. Our focus is on the size distribution of clones after some time, which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. We prove that the large time limit of the clone size distribution has a general two-parameter form for a large class of population growth. The large time clone size distribution always has a power-law tail, and for subexponential wild-type growth the probability of a given clone size is inversely proportional to the clone size. We support our results by analysing a data-set on tumour metastasis sizes, and we find that a power-law tail is more likely than an exponential one, in agreement with our predictions.

Link: http://arxiv.org/abs/1604.04936#

Stochasticity in the Genotype-Phenotype Map: Implications for the Robustness and Persistence of Bet-Hedging

Stochasticity in the Genotype-Phenotype Map: Implications for the Robustness and Persistence of Bet-Hedging

Daniel Nichol, Mark Robertson-Tessi, Peter Jeavons, Alexander RA Anderson

Abstract

For the last few decades modern biology has focused on quantifying, understanding and mapping the genetic characteristics of cells. This genotype-driven perspective has led to significant advances in our understanding and treatment of diseases such as cancer e.g. the discovery of driver mutations and the development of molecularly-targeted therapeutics. However, this perspective has largely ignored the functional outcome of genetic changes: the cellular phenotype. In part, this is simply because phenotypes are neither easy to define or measure as they critically depend on both genotype and context. Heterogeneity at the gene scale has been known for sometime, and there has been significant effort invested in trying to find patterns within it, but much less is understood about how this heterogeneity manifests itself in phenotypic change, i.e. the genotype-phenotype map (GP-map). This mapping is not one-to-one but many-to-many and is fundamentally the junction at which both genes and environment meet to produce phenotypes. Many genotypes produce similar phenotypes, and multiple phenotypes can emerge from a single genotype. To further complicate matters, genetically identical cells in uniform environments still exhibit phenotypic heterogeneity. Therefore a central open question in biology today is how can we connect the abundance of genomic data with cell phenotypic behaviour, this is especially pertinent to the issue of treatment resistance as many therapies act on cellular phenotypes. Our focus here is to tackle the GP-map question through the use of the simplest functional mapping we can define that also captures phenotypic heterogeneity: a molecular switch. Molecular switches are ubiquitous in biology, observed in many organisms and naturally map molecular components to decisions (i.e. phenotypes). Often stochastic in nature, such switches can be the difference between life or death in environments that fluctuate unpredictably, since they will ensure that at least some offspring are adapted to future environments. For convenience we use Chemical Reaction Networks (CRNs) to define the map of gene products to phenotypes, allowing us to investigate the impact of distinct mappings (CRNs) and perturbations to them. We observe that key biological properties naturally emerge, including both robustness and persistence. Robustness may explain why such bet hedging strategies are common in biology, and not readily destroyed through mutation. Whereas persistence may explain the apparent paradox of bet-hedging - why does phenotypic hedging exist in environments beneficial to only one of the phenotypes, when selection necessarily acts against it? The structure of the molecular switch, itself subject to selection, can slow the loss of hedging to ensure a survival mechanism even against environmental catastrophes which are very rare. Critically, these properties when taken together have profound and significant implications for the emergence of treatment resistance, since the timescale of extinction depends heavily on the underlying GP-map.

doi: http://dx.doi.org/10.1101/042424

A stochastic individual-based model for immunotherapy of cancer

Martina Baar, Loren Coquille, Hannah Mayer, Michael Hölzel, Meri Rogava, Thomas Tüting, Anton Bovier 

 

We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modelling of immunotherapy for malignant tumors. The main characteristics of the model are distinguishing phenotype and genotype, including environment-dependent transitions between phenotypes that do not affect the genotype, and the introduction of a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. We prove that this stochastic process converges in the limit of large populations to a deterministic limit which is the solution to a system of quadratic differential equations. We illustrate the new setup by using it to model various phenomena arising in immunotherapy. Our aim is twofold: on the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events (which cannot be obtained with deterministic systems) may help to understand the resistance of tumors to various therapeutic approaches and may have non-trivial consequences on tumor treatment protocols and demonstrate this through numerical simulations.

http://arxiv.org/abs/1505.00452 

Stochastic model for computer simulation of the number of cancer cells and lymphocytes in homogeneous sections of cancer tumors

Arnulfo Castellanos-Moreno, Alejandro Castellanos-Jaramillo, Adalberto Corella-Madueño, Sergio Gutiérrez-López, Rodrigo Rosas-Burgos

(Submitted on 14 Oct 2014)

We deal with a small enough tumor section to consider it homogeneous, such that populations of lymphocytes and cancer cells are independent of spatial coordinates. A stochastic model based in one step processes is developed to take into account natural birth and death rates. Other rates are also introduced to consider medical treatment: natural birth rate of lymphocytes and cancer cells; induced death rate of cancer cells due to self-competition, and other ones caused by the activated lymphocytes acting on cancer cells. Additionally, a death rate of cancer cells due to induced apoptosis is considered. Weakness due to the advance of sickness is considered by introducing a lymphocytes death rate proportional to proliferation of cancer cells.
Simulation is developed considering different combinations of the parameters and its values, so that several strategies are taken into account to study the effect of anti-angiogenic drugs as well the self-competition between cancer cells. Immune response, with the presence of a kind of specialized lymphocytes, is introduced such that they appear once cancer cells are detected. Induced apoptosis of cancer cells is introduced to model the action of several drugs under development right now. Besides, the model predicts the cancer relapse even from a very small number of cells. Simulation is done by using Gillespie algorithm.

link

A tug-of-war between driver and passenger mutations in cancer and other adaptive processes

Christopher Dennis McFarland, Leonid A Mirny, Kirill S Korolev

doi: 10.1101/003053

Abstract

Cancer progression is an example of a rapid adaptive process where evolving new traits is essential for survival and requires a high mutation rate. Precancerous cells acquire a few key mutations that drive rapid population growth and carcinogenesis. Cancer genomics demonstrates that these few ‘driver’ mutations occur alongside thousands of random ‘passenger’ mutations––a natural consequence of cancer's elevated mutation rate. Some passengers can be deleterious to cancer cells, yet have been largely ignored in cancer research. In population genetics, however, the accumulation of mildly deleterious mutations has been shown to cause population meltdown. Here we develop a stochastic population model where beneficial drivers engage in a tug-of-war with frequent mildly deleterious passengers. These passengers present a barrier to cancer progression that is described by a critical population size, below which most lesions fail to progress, and a critical mutation rate, above which cancers meltdown. We find support for the model in cancer age-incidence and cancer genomics data that also allow us to estimate the fitness advantage of drivers and fitness costs of passengers. We identify two regimes of adaptive evolutionary dynamics and use these regimes to rationalize successes and failures of different treatment strategies. We find that a tumor’s load of deleterious passengers can explain previously paradoxical treatment outcomes and suggest that it could potentially serve as a biomarker of response to mutagenic therapies. The collective deleterious effect of passengers is currently an unexploited therapeutic target. We discuss how their effects might be exacerbated by both current and future therapies.

From the bioRxiv :  http://biorxiv.org/content/early/2014/02/26/003053

Investigating the relation between stochastic differentiation and homeostasis in intestinal crypts via multiscale modeling

Alex Graudenzi, Giulio Caravagna, Giovanni De Matteis, Marco Antoniotti

doi: 10.1101/000927

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ABSTRACT

Colorectal tumors originate and develop within intestinal crypts. Even though some of the essential phenomena that characterize crypt structure and dynamics have been effectively described in the past, the relation between the differentiation process and the overall crypt homeostasis is still partially understood. We here investigate this relation and other important biological phenomena by introducing a novel multiscale model that combines a morphological description of the crypt with a gene regulation model: the emergent dynamical behavior of the underlying gene regulatory network drives cell growth and differentiation processes, linking the two distinct spatio-temporal levels. The model relies on a few a priori assumptions, yet accounting for several key processes related to crypt functioning, such as: dynamic gene activation patterns, stochastic differentiation, signaling pathways ruling cell adhesion properties, cell displacement, cell growth, mitosis, apoptosis and the presence of biological noise. We show that this modeling approach captures the major dynamical phenomena that characterize the regular physiology of crypts, such as cell sorting, coordinate migration, dynamic turnover, stem cell niche maintenance and clonal expansion. All in all, the model suggests that the process of stochastic differentiation might be sufficient to drive the crypt to homeostasis, under certain crypt configurations. Besides, our approach allows to make precise quantitative inferences that, when possible, were matched to the current biological knowledge and it permits to investigate the role of gene-level perturbations, with reference to cancer development. We also remark the theoretical framework is general and may applied to different tissues, organs or organisms.

link: http://www.biorxiv.org/content/early/2013/11/25/000927

Accounting for Intrinsic and Extrinsic Randomness in Studying Cancer Cell Population Dynamics

Is cancer growth random and stochastic or ordered and deterministic? Are these ideas mutually exclusive?

Siavash Ghavami, Olaf Wolkenhauer, Farshad Lahouti, Mukhtar Ullah, Michael Linnebacher

(Submitted on 8 Oct 2013)

Studying the development of malignant tumours, it is important to know and predict the proportions of different cell types in tissue samples. Knowing the expected temporal evolution of the proportion of normal tissue cells, compared to stem-like and non-stem like cancer cells, gives an indication about the progression of the disease and indicates the expected response to interventions with drugs. Such processes have been modeled using Markov processes. An essential step for the simulation of such models is then the determination of state transition probabilities. We here consider the experimentally more realistic scenario in which the measurement of cell population sizes is noisy, leading to a hidden Markov model. In this context, extrinsic randomness is related to noisy measurements, which are used for the estimation of the transition probability matrix. Intrinsic randomness, on the other hand, is here related to the error in estimating the state probability from small cell populations. Using aggregated data of fluorescence-activated cell sorting (FACS) measurement, we develop a minimum mean square error estimator (MMSE) and maximum likelihood (ML) estimator and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes using a transition probability matrix estimated from noisy data. We analyze the properties of two estimators for different noise distributions and prove an optimal solution for Gaussian distributions with the MMSE. Our numerical results show, that for noisy measurements the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach in which measurements are assumed to be noise free.

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

Simon Holbek, Kristian Moss Bendtsen, Jeppe Juul

(Submitted on 18 May 2013)

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 life span 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 favour a shorter than optimal average cancer free life span in order to postpone cancer onset until late in life.

 Link to the arXiv.