Cancer Dynamics

In this theme we will address a number of biological and mathematical issues related to modeling of evolution of cancer, organized in three core lectures, which will cover the fundamental issues of cell proliferation and mutation dynamics, molecular events affecting specific pathways in cells and the population genetics effects (see the abstracts further on)

This section of the summer school will include a number of instructor-suggested group projects, in which students will apply various numerical techniques to formulate, identify and solve stochastic models of cancer evolution. Students will then apply these tools to model experimental and clinical data. This section of the summer school is organized by Marek Kimmel. Please address all questions about this section of the summer school to its organizer.

Core Instructors

• Rosemary Braun, Northwestern, rbraun@northwestern.edu
• Alexandra Jilkine, Notre Dame, Alexandra.Jilkine.1@nd.edu
• Marek Kimmel, Rice, kimmel@rice.edu (Course Leader)
• Natalia Komarova, UC Irvine, komarova@uci.edu

Course-specific Instructors

• Seth Corey, Northwestern U
• Vittorio Cristini, UNM
• Nicolas Flores, Rice
• Tomasz Lipniacki, IPPT, Warsaw, Poland
• Dominik Wodarz, UC Irvine

Core lectures

Inferring Aberrant Network Dynamics in Cancer Cells

Rosemary Braun, Northwestern University, rbraun@northwestern.edu

Contents

Cellular processes are coordinated by complex networks of regulatory interactions. Alterations of specific network elements (such as differential expression of a gene in the pathway) may have varying severity depending on the surrounding network structure; while some may have profound consequences on the regulatory dynamics, others may be tolerated or compensated by adaptive changes to other network elements. Contemporary pathway analyses of gene expression profiling data, however, often searches for pathways that exhibit an overrepresentation of differentially expressed genes, without considering the role of those genes in the network and without distinguishing between compensatory or cumulative alterations. I will discuss a novel graph-theoretic method for pathway analysis of transcription profiling data that takes into account the network structure. The method can identify pathways that appear to be differentially regulated even when the proportion of differentially expressed genes is small, and, relatedly, pathways that are expected to maintain their function despite many differentially expressed genes (suggesting compensatory adaptations). We apply this method to gene expression profiling data from GCSFR mutant and wild-type cells to investigate the aberrant GCSF signaling mechanisms driving the progression from severe congenital neutropenia to acute myeloid leukemia.

Stochastic models of stem cell renewal and dedifferentiation in cancer

Alexandra Jilkine, University of Notre Dame, Alexandra.Jilkine.1@nd.edu

Contents

Recent evidence suggests that, like many normal tissues, cancers are often maintained by a small population of cancer stem cells that divide indefinitely to produce more differentiated cancerous cells. Tissues, however, contain many more differentiated cells than stem cells, and mutations may cause such cells to "dedifferentiate" into a stem-­‐like state. I will review some commonly used models of stem cell renewal and homeostasis, and its regulation by various potential negative feedback loops. Looking at the effects of dedifferentiation on the time to cancer onset, I found that the effect of dedifferentiation depends critically on how stem cell numbers are controlled by the body. If homeostasis is very tight (due to all divisions being asymmetric), then dedifferentiation has little effect, but if homeostatic control is looser (allowing both symmetric and asymmetric divisions), then dedifferentiation can dramatically hasten cancer onset and lead to exponential growth of the cancer stem cell population, even if homeostasis is maintained. We consider both space-­‐free and spatial versions of this process to look at effect that tissue architecture can play in this process.

Modeling Evolution of the Myelodysplastic Syndrome

Marek Kimmel, Rice University, kimmel@rice.edu

Contents

We present a model of evolution of the myelodysplastic syndrome arising as intermediate stage between severe congenital neutropenia and acute lymphoid leukemia, when SCN is treated by massive doses of the granulocyte colony-stimulating factor. The dominant (75%) driver mutation in MDS patients is the D715 truncation of the GCSF receptor. We consider modified Moran model with directional selection to compute time to fixation of D175 in the population. Selective advantage maybe attributable to a D715-related perturbation of the STAT/SOCS circuit in granulocyte precursors leading to altered proliferation and differentiation; we present some experimental and model-based evidence for this. Finally, using a stochastic model, we discuss the hypothesis that of a compensatory mechanism counteracting the effects of ELAINE mutations is active before the GCSF treatment is initiated. The SCN --> MDS --> AML model is important as a contribution to understanding leukemogenesis.

Cooperation, space, and evolution in cancer

Natalia Komarova, UC Irvine, komarova@uci.edu

Contents

Cancer comes about by a sequence of mutations that change the cells' fitness and create advantageous phenotypes. These phenotypes displace other cells and spread, thus winning the evolutionary competition. It is possible that in order to create those advantageous mutants, several different mutations have to be accumulated in a cell, such that each individual mutation is disadvantageous, and together they comprise a fitness advantage. In the literature, this is often called "crossing a fitness valley". In this talk I will discuss two mechanisms that can accelerate fitness valley crossing. One is connected with spatial interactions of cells. Space can have a very complex role in fitness valley crossing, and can accelerate or decelerate evolution. The second mechanism involves the notion of cooperation among the cells, where shared benefits are received through "division of labor". I will show how in such context, cooperation can speed up the evolutionary process. Moreover, the emergence of cheaters that destroy cooperation dynamics can “unite” all mutations within one individual on a fast time scale. Paradoxically, the presence of such cheaters happens to accelerate evolution even more.

Course-specific lecture sequences

Bioinformatics of Cancer Networks

Rosemary Braun, Northwestern University, rbraun@northwestern.edu

The cellular proliferation, migration, and invasion characteristics that are the hallmarks of cancer are due to aberrant signaling in the regulatory networks that ordinarily control growth and apoptosis. These pathways can be compromised in a variety of ways, both in terms of the affected molecules and in terms of the mechanism (eg, by mutation or by altered transcription). Today, modern high-throughput assays yield genome-wide profiles of sequence variation, transcription factor binding, methylation, and expression for each sample of interest, and this exquisitely detailed information provides an unprecedented opportunity to characterize the molecular mechanisms governing malignant transformation. At the same time, the high dimensionality of the data presents analytical challenges. Mathematical models of regulatory networks are essential for identifying pathological signaling processes in cancer cells. In these lectures, we will discuss various approaches for the systems-level analysis of high-throughput data.

• Lecture 1:

Introduction to basic statistical and bioinformatic analysis of genome-wide profiling data, including popular non-network approaches; incorporating pathway network topology into the analysis --introduction to graph theory, approaches to analyze experimental data in the context of networks derived from expert-knowledge pathway databases; comparison of methods & discussion of open challenges;

• Lecture 2:

Inferring network topology: methods for reconstructing regulatory network structure from experimental data, including an overview of probabilistic graphical models, and graph-theoretic (SPaTO, PDM) and information-theoretic (ARACNe) network inference techniques.

Crash Course in Stochastic Processes

Nicolas Flores Castillo, Rice University, nicolas@rice.edu

Contents

This will be an introductory course to Applied Probability and Stochastic Processes. The course will cover essentially two parts: The first part will be an introduction to the foundations of probability theory and the second part will be about applications of probability that are commonly used in engineering, statistical and population genetics. More precisely, these applications or tools are discrete-time and continuous-time stochastic processes. Time permitting, we will also discuss some more advanced topics. Some of the topics that might be discussed are: axioms of probability theory, conditional probability, independence, discrete and continuous random variables, expectation, standard discrete and continuous distribution functions, transform techniques, modes of convergence of random variables, central limit theorems, the Poisson process, an introduction to discrete-time Markov chains (including recurrence, transience and absorbing states), and branching processes.

Course objectives:

• Learn probability vocabulary and incorporate the probability knowledge to real life problems
• Understand the differences between deterministic and probabilistic models
• Know the advantages and the limitations of some probability models
• Provide foundations for applying probability theory in mathematical statistics and stochastic processes
• Exploit the flexibility of stochastic models
• Recognize the potential of asymptotic theory to make predictions or inferences about a specific population of individuals

Textbook: I will provide notes of my own which will be based on the following books:

• Olofsson: Probability, Statistics, and Stochastic Processes
• Ross: Introduction to Probability Models, 10th Edition, 2010 Academic Press.
• Hogg and Tanis: Probability and Statistical Inference
• Ross: A First Course in Probability
• Kulkarni: Introduction to Modeling and Analysis of Stochastic Systems

Other useful references:

• Grimmett: Probability and Random Processes
• Resnick: Adventures in Stochastic Processes
• Karlin and Taylor: A first course in Stochastic Processes

Mathematical Models of Mutation Acquisition and Time to Cancer

Alexandra Jilkine, Notre Dame University, Alexandra.Jilkine.1@nd.edu

Contents

I will cover birth-­‐death processes, and their applications to modeling populations of cancerous cells. Models from population genetics, emphasizing phenomena of genetic drift and clonal extinction, will be used. Effect of fitness and selection on evolutionary dynamics will be covered. These types of models can be used to reconstruct the order in which oncogenic mutations arise, and most likely path to mutation acquisition. Impact of deleterious passengers on cancer progression will be mentioned.

References: Textbooks:

• Linda Allen. An Introduction to Stochastic Processes with Applications to Biology. CRC press, 2nd edition.
• Weinberg, Robert. The biology of Cancer. Garland Science, 2nd edition.

Other references

• Bozic, Ivana, Tibor Antal, Hisashi Ohtsuki, Hannah Carter, Dewey Kim, Sining Chen, Rachel Karchin, Kenneth W. Kinzler, Bert Vogelstein, and Martin A. Nowak. "Accumulation of driver and passenger mutations during tumor progression." Proceedings of the National Academy of Sciences 107, no. 43 (2010): 18545-­‐18550.
• McFarland CD, Korolev KS, Kryukov GV, Sunyaev S, Mirny LA The impact of deleterious passenger mutations on cancer progression. PNAS 2013 110(7).
• Ramon Diaz-­‐Uriarte .Inferring restrictions in the temporal order of mutations during tumor progression: effects of passenger mutations, evolutionary models, and sampling. Click

Modeling cell cycle kinetics of normal and cancer cells and anti-cancer therapy

Marek Kimmel, Rice University, kimmel@rice.edu

Contents

• Lecture 1: Basics of cell cycle kinetics. Deterministic (ODE) and stochastic (branching process) models of cell cycle and cell proliferation. Clonal theory of drug resistance. Estimation of parameters of cell cycle.
• Lecture 2: Modeling chemotherapy in vivo. Optimization of therapy using optimal control methods. Drug resistance/gene amplification model using branching random walk.
• Lecture 3: Modeling chemotherapy and resistance in vivo. Michor’s model of leukemia therapy. Role of stem cells in chemotherapy (Stiehl’s et al. model). Early detection of lung cancer, therapy and survival (Goldwasser and Kimmel study).

References (idiosyncratic and ad hoc)

• Goldwasser, Deborah L., and Marek Kimmel. "Small median tumor diameter at cure threshold (< 20 mm) among aggressive non‐small cell lung cancers in male smokers predicts both chest X‐ray and CT screening outcomes in a novel simulation framework." International Journal of Cancer 132.1 (2013): 189-197.
• Kimmel, Marek, and Andrzej Swierniak. "Control theory approach to cancer chemotherapy: Benefiting from phase dependence and overcoming drug resistance." Tutorials in Mathematical Biosciences III. Springer Berlin Heidelberg, 2006. 185-221.
• Kimmel, Marek, and David E. Axelrod. "Branching Processes in Biology. Interdisciplinary Applied Mathematics 19." (2002).
• Michor, Franziska, et al. "Dynamics of chronic myeloid leukaemia." Nature 435.7046 (2005): 1267-1270.
• Ortiz-Tudela, E., A. Mteyrek, A. Ballesta, P. F. Innominato, and F. Lévi. "Cancer chronotherapeutics: experimental, theoretical, and clinical aspects." In Circadian clocks, pp. 261-288. Springer Berlin Heidelberg, 2013.
• Swierniak, A., A. Polanski, and M. Kimmel. "Optimal control problems arising in cell‐cycle‐specific cancer chemotherapy." Cell proliferation 29.3 (1996): 117-139.
• Świerniak, A., et al. "Qualitative analysis of controlled drug resistance model-inverse Laplace and and semigroup approach." Control and Cybernetics 28 (1999): 61-73.
• Swierniak, Andrzej, Marek Kimmel, and Jaroslaw Smieja. "Mathematical modeling as a tool for planning anticancer therapy." European journal of pharmacology 625.1 (2009): 108-121.

Deterministic and stochastic modeling of cancer

Natalia Komarova, UC Irvine, komarova@uci.edu

Contents

• Lecture 1: Deterministic modeling. Short introduction into aspects of cancer. Simple deterministic modeling. Exponential, logistic, and other growth laws for tumors. Two-species growth and competition. Axiomatic modeling. Angiogenesis, inbibitors and promoters in cancer growth.
• Lecture 2. Deterministic modeling continued. Quasispecies equations. Proliferation cascades. Mutator phenotype: stable and unstable cells. Microsatellite instability and genetic instability.
• Lecture 3. Stochastic modeling. Basic facts about chronic myeloid leukemia. Birth death processes with mutations and modeling of leukemia. Probability generating function and the method of characteristics. Treatment with one and two drugs.
• Lecture 4. Stochastic modeling continued. Moran process and cancer initiation. Oncogenes and gain-of-function mutations. Probability of mutant fixation. Tumor suppressor genes and loss-of-function mutations. Stochastic tunneling.

References:

• Wodarz and Komarova. (2014) Dynamics of cancer: Mathematical foundations in oncology. World Scientific.
• Komarova and Wodarz (2013) Targeted Cancer Treatment in Silico: Small Molecule Inhibitors and Oncolytic Viruses (Modeling and Simulation in Science, Engineering and Technology). Birkhauser.

Course-specific lectures

• Seth Corey, Northwestern
• Vittorio Cristini, U New Mexico

Feedbacks, delays, and bifurcation structures of cancer related regulatory systems and their biological interpretations

• Tomasz Lipniacki, IPPT Warsaw, Poland

I will discuss three regulatory systems of NF-κB, p53 and MAPK analyzing correspondence between structures of regulatory networks and emerging dynamics. The regulatory elements responsible for particular bifurcations will be identified, and the biological interpretation of the obtained bifurcation diagrams will be given. In the case of p53, emergence of (a bit exotic) Neimark-Sacker bifurcation allows for behavior in which the only two stable recurrent solutions are the limit cycle oscillations of p53 level, and the stable steady state corresponding to a very high level of p53 resulting in apoptosis. Using the example of the NF-κB system I will show the correspondence between bifurcation structure obtained for deterministic approximation and stochastic dynamics exhibited by single cells and simulated using Gillespie algorithm. In the case of MAPK signaling I will analyze the correspondence between dynamics obtained for ordinary differential equations (relaxation oscillations) and dynamics exhibited by related partial differential equations (local excitation -- global inhibition).

Mathematical models of cancer treatment with oncolytic viruses

• Dominik Wodarz, UC Irvine

Oncolytic viruses are viruses that specifically infect cancer cells, replicate in them, kill them, and spread to further cancer cells. They have been used as a treatment approach against cancers in a variety of clinical trials. While promising results have been observed in clinical settings, consistent success remains elusive. Mathematical models provide an important avenue of research in the design of such therapies. They allow us to explore the conditions that likely lead to virus-mediated cancer control, and the conditions that likely lead to failure. I will explore different kinds of mathematical and computations models, ranging from simple to complex, from non-spatial to spatial models. Biological insights, as well as mathematical/computational challenges will be discussed. In particular, I will explore how different mathematical formulations of certain biological processes can significantly influence the dynamics and the outcome, and discuss the issue of robustness of results that arise from mathematical models.

Projects

Network properties and cancer association (Braun)

In these projects, we seek to address whether the network properties of a gene are related to its role in cancer. We know already that genes differentially expressed in lung cancer also happen to have high centrality in the interactome (doi:10.1093/bioinformatics/bti688) and that a selection of ~350 cancer-related genes tend to have greater degrees than other genes in the PPI (doi:10.1093/bioinformatics/btl390). However, most such studies consider only one modality of data (ie, only expression, only CNV, only SNP, etc.) and/or only one disease (eg, lung cancer specifically). Because there may be more than one way to deal a deleterious blow to a pathway, an integrative analysis is of greater interest. In addition, it would be useful to know if there are particular patters that are associated with specific phenotypes vs. being a global feature of cancers (or even of all disease!). Using data from the TCGA project, it is possible to address questions such as:

• A.1) Do we find non-random associations between vertex/edge properties and the significance of genes to the phenotype of interest?

Are these patterns consistent across all cancers? Other diseases? Do we find that as we integrate data sets for a particular disease, those relationships become stronger (ie, are nodes more likely to be hit by a UNION of DE, mutation, methylation, miRNA targeting, &c. when they are highly central than not)? What about edge properties? Do we find that high-centrality edges are more likely to exhibit differential co-expression?

• A.2) Does network structure confer robustness?

Assortativity of "hit" genes on the network may imply that the network structure itself confers robustness to random hits (ie, for the alterations to cause disease, they must occur in a way that is non-randomly associated with the network structure itself). Do we find that pairs of genes that are "hit" (by differential expression or by mutation or by ...) are more likely to have an edge connecting them than expected by chance? (This may be thought of as evidence for Knudsen's famous "two hit" hypothesis at the network level!) Alternatively, do we find that genes with high clustering coefficients are less likely to disease-associated variation than ones with low clustering coefficients?

• A.3) Identifying "hubs" of dysregulation from time-course data.

Genes (nodes) or interactions (edges) that play a crucial biological role may, when altered, lead to a "wave" of differential expression spreading outward across the network over time. Identifying these hubs may help distinguish between differentially expressed genes that are drive the phenotype from those for which differential expression is a response to the drivers. Using time-course expression data from wildtype and mutant cell lines, we will identify centers of waves of differential expression that spread across the network.

Inferring novel interactions (Braun)

• B.1) micro-RNAs [miRNAs] are short (10-20bp) non-coding RNA sequences that bind to complementary mRNA targets and inhibit translation. As with protein-coding genes, miRNA profiling has led to the generation of lists of differentially expressed miRNA signatures associated with cancer. However, because a given mRNA may be targeted by many miRNAs, and because a given miRNA may target multiple mRNAs, the mechanistic significance of the differential expression of a set of miRNAs is not necessarily clear. Using TCGA data and an approach similar to the gene--pathway pair analysis described in (doi:10.1186/1471-2105-9-488), can we identify miRNAs that appear to have pathway-wide regulatory effects?

• B.2) Detecting network re-wiring: can we find "new" or "missing" edges in cancer cells?

We begin by searching for differential co-expression (eg, differential gene-gene correlation) in tumor vs. normal cells. Using these correlation differences and putative pathway networks (which should ideally be representative of the normal biology), the goal is to identify "cut" edges (gene-pairs connected by edges that exhibit lower co-expression in tumor) and "new" edges (gene-pairs that have higher co-expression in tumor but are not represented by an edge). Secondarily, one may also ask whether the flow of a signal through the "rewired" tumor network is significantly different from normal network by comparing the spectra of the two graphs.

• B.3) Inferring regulatory networks using probabilistic models of time-course data.

Here, we will attempt to reconstruct gene regulatory networks by modeling gene expression as a mixture of gamma distributions (representing "on" and "off", respectively), allowing us to assign to each gene a probability of being in an "up"/"on" state (eg, as in doi:10.1371/journal.pone.0000425) over time. From these probabilities, we compute the mutual information between gene pairs; by adaptively selecting a threshold at which mutual information is considered significant evidence for an edge, we obtain a sparse network of high MI edges. The inferred networks may then be compared to known pathways.

Further exploration of the cancer stem cell hypothesis and inter-­‐conversion to non-­‐CSC cells (Jilkine)

Extend the simple Markov Chain model by Gupta et al. that explores how non-­‐cancer stem cells (NCSCs) can change into cancer stem cells (CSCs) and maintain a constant proportion of the equilibrium proportion of CSCs. The model only includes inter-­‐conversion, but can be made more realistic by including the effects of cell division and population growth. Suppose that the CSCs can also have two states: non-­‐dividing, quiescent cells and proliferating cells. The proliferating cells can give rise to more cancer stem cells or more differentiated cells that can divide a given number of times before dying. Consider cases where the dynamics of the cancer stem cell population is independent of the differentiated cells, as well as cases where the probabilities of transitioning between quiescent and proliferating states depend on the size of the tumor. Consider effects of both inhibitory and stimulatory signals from the tumor. Suppose a drug targets only the rapidly dividing cells. What will happen to the proportion of CSCs in the tumor over time? How long before emergence of drug resistance in the proliferating population? Some recent papers to consider:

• Gupta et al. Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells. Cell. 2011 Aug 19;146(4):633-­‐44.
• Li L, Clevers H. Coexistence of quiescent and active adult stem cells in mammals. Science. 2010 Jan 29;327(5965):542-­‐5.
• Liu et al. Nonlinear Growth Kinetics of Breast Cancer Stem Cells: Implications for Cancer Stem Cell Targeted Therapy. Sci. Rep. 3, 2473; DOI:10.1038/ srep02473 (2013).

Modeling metastasis (Jilkine)

Estimating the probability of rare events driving cancer metastasis is still an open problem. Given the long time-­‐scale of most metastatic disease, it is commonly presumed that colonization arises from rare founding cells that are pre-­‐adapted for colonizing new environments. Use the multi-­‐type branching process of the type used in Danesh et al. to calculate the time of spread (see diagram 6 in Scott et al for an idea of how to model the organ network). Some recent papers to consider:

• Danesh K, Durrett R, Havrilesky LJ, Myers E. A branching process model of ovarian cancer. Theor Biol. 2012 Dec 7;314:10
• Scott JG, Gerlee P, Basanta D, Fletcher AG, Maini PK, Anderson AR. Mathematical Modeling of the Metastatic Process, in Experimental Metastasis: Modeling and Analysis, ed. A. Malek (2013) Springer: New York, Chapter 9, 189-­‐ 208.http://arxiv.org/abs/1305.4622
• Vanharanta and Massague. Origins of Metastatic Traits. Cancer Cell. Oct 14, 2013; 24(4): 410–421.

Chemotherapy in vitro model (Kimmel)

Set up a computational model of cell cycle kinetics in the form of a system of ODEs or any other way you develop or identify in the literature. Construct a model of chemotherapy using one or two agents with differing cell-cycle specificity. Suggest optimal control and/or heuristic therapy protocols and dosing a computational experiment to evaluate their action.

Emergence of drug resistance (Kimmel)

Identify in the literature a mechanism of cell resistance to an anticancer agent. Based on a model of cell proliferation, and (if feasible) a molecular-level model of drug action, devise a model of treatment in vitro or in vivo. Examine emergence of drug resistance in the model and compare the outcome with experimental or clinical data.

Studying stem cell strategies (Komarova)

Stem cells are capable of two types of divisions: symmetric and asymmetric. In this project we will calculate an optimal strategy for stem cells to maximally delay the onset of cancer, in a spatial setting of stem cell niches.

References:

• Shahriyari, L., & Komarova, N. L. (2013). Symmetric vs. Asymmetric Stem Cell Divisions: An Adaptation against Cancer?. PloS one, 8(10), e76195.

Cell motility as a vehicle of microevolution (Komarova)

It has been shown previously that spatial interactions can have an interesting and nontrivial influence on the speed of evolution, especially in the context of tumor suppressor gene mutations. We will investigate how cell motility can speed up cancer progression.

References:

• Komarova, N. L., Shahriyari, L., & Wodarz, D. (2014). Complex role of space in the crossing of fitness valleys by asexual populations. Journal of The Royal Society Interface, 11(95), 20140014.