Mutual Hazard Networks: Markov chain models of cancer progression

Cancer progresses by accumulating genomic events, such as mutations and copy number alterations, whose chronological order is key to understanding the disease but difficult to observe. Instead, cancer progression models use co-occurrence patterns in cross-sectional data to infer dependencies between events and thereby uncover their most likely order of occurrence. Here we introduce Mutual Hazard Networks, a new class of models that improve upon the state of the art by allowing stochastic dependencies between events, inhibiting dependencies and dependencies that form cycles. MHNs model events by their spontaneous rate of occurrence and by multiplicative effects they exert on the rates of successive events. We further propose an approach for modeling and predicting pivotal events such as the diagnosis of the tumor, temporary inflammation of the tumor, seeding of a metastasis or death of the patient. To this end we formulate an MHN as a large, continuous-time Markov chain whose transition rate matrix is given as a sum of tensor products. We develop efficient algorithms for computing its transient, stationary and time-marginal probability distributions as well as their derivatives with respect to model parameters in order to perform inference. First results indicate that MHNs consistently outperform state-of-the-art models on publicly available data in terms of cross-validated model fit. In particular, MHN inferred from a glioblastoma dataset from The Cancer Genome Atlas that IDH1 mutations are early events that promote subsequent mutations in TP53, a finding that is independently supported by consecutive biopsies. Moreover, we demonstrate the general usefulness of our method beyond oncology by applying it to a stochastic SIR model of epidemic spread. We use our algorithm for computing the derivative of a transient distribution to estimate the monthly infection and recovery rates during the first COVID-19 wave in Austria in a full Bayesian analysis

Modelling cancer progression using Mutual Hazard Networks

Motivation: Cancer progresses by accumulating genomic events, such as mutations and copy number alterations, whose chronological order is key to understanding the disease but difficult to observe. Instead, cancer progression models use co-occurence patterns in cross-sectional data to infer epistatic interactions between events and thereby uncover their most likely order of occurence. State-of-the-art progression models, however, are limited by mathematical tractability and only allow events to interact in directed acyclic graphs, to promote but not inhibit subsequent events, or to be mutually exclusive in distinct groups that cannot overlap. Results: Here we propose Mutual Hazard Networks (MHN), a new Machine Learning algorithm to infer cyclic progression models from cross-sectional data. MHN model events by their spontaneous rate of fixation and by multiplicative effects they exert on the rates of successive events. MHN compared favourably to acyclic models in cross-validated model fit on four datasets tested. In application to the glioblastoma dataset from The Cancer Genome Atlas, MHN proposed a novel interaction in line with consecutive biopsies: IDH1 mutations are early events that promote subsequent fixation of TP53 mutations. Availability Implementation and data are available at https://github.com/RudiSchill/MHN

Low-rank tensor methods for Markov chains with applications to tumor progression models

Cancer progression can be described by continuous-time Markov chains whose state space grows exponentially in the number of somatic mutations. The age of a tumor at diagnosis is typically unknown. Therefore, the quantity of interest is the time-marginal distribution over all possible genotypes of tumors, defined as the transient distribution integrated over an exponentially distributed observation time. It can be obtained as the solution of a large linear system. However, the sheer size of this system renders classical solvers infeasible. We consider Markov chains whose transition rates are separable functions, allowing for an efficient low-rank tensor representation of the linear system’s operator. Thus we can reduce the computational complexity from exponential to linear. We derive a convergent iterative method using low-rank formats whose result satisfies the normalization constraint of a distribution. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank

Mechanisms governing the pioneering and redistribution capabilities of the non-classical pioneer PU.1

Establishing gene regulatory networks during differentiation or reprogramming requires master or pioneer transcription factors (TFs) such as PU.1, a prototype master TF of hematopoietic lineage differentiation. To systematically determine molecular features that control its activity, here we analyze DNA-binding in vitro and genome-wide in vivo across different cell types with native or ectopic PU.1 expression. Although PU.1, in contrast to classical pioneer factors, is unable to access nucleosomal target sites in vitro, ectopic induction of PU.1 leads to the extensive remodeling of chromatin and redistribution of partner TFs. De novo chromatin access, stable binding, and redistribution of partner TFs both require PU.1's N-terminal acidic activation domain and its ability to recruit SWI/SNF remodeling complexes, suggesting that the latter may collect and distribute co-associated TFs in conjunction with the non-classical pioneer TF PU.1

Modelling cancer progression using Mutual Hazard Networks

Motivation Cancer progresses by accumulating genomic events, such as mutations and copy number alterations, whose chronological order is key to understanding the disease but difficult to observe. Instead, cancer progression models use co-occurrence patterns in cross-sectional data to infer epistatic interactions between events and thereby uncover their most likely order of occurrence. State-of-the-art progression models, however, are limited by mathematical tractability and only allow events to interact in directed acyclic graphs, to promote but not inhibit subsequent events, or to be mutually exclusive in distinct groups that cannot overlap. Results Here we propose Mutual Hazard Networks (MHN), a new Machine Learning algorithm to infer cyclic progression models from cross-sectional data. MHN model events by their spontaneous rate of fixation and by multiplicative effects they exert on the rates of successive events. MHN compared favourably to acyclic models in cross-validated model fit on four datasets tested. In application to the glioblastoma dataset from The Cancer Genome Atlas, MHN proposed a novel interaction in line with consecutive biopsies: IDH1 mutations are early events that promote subsequent fixation of TP53 mutations. Availability and implementation Implementation and data are available at https://github.com/RudiSchill/MHN. Supplementary information Supplementary data are available at Bioinformatics online

Low-rank tensor methods for Markov chains with applications to tumor progression models

Continuous-time Markov chains describing interacting processes exhibit a state space that grows exponentially in the number of processes. This state-space explosion renders the computation or storage of the time-marginal distribution, which is defined as the solution of a certain linear system, infeasible using classical methods. We consider Markov chains whose transition rates are separable functions, which allows for an efficient low-rank tensor representation of the operator of this linear system. Typically, the right-hand side also has low-rank structure, and thus we can reduce the cost for computation and storage from exponential to linear. Previously known iterative methods also allow for low-rank approximations of the solution but are unable to guarantee that its entries sum up to one as required for a probability distribution. We derive a convergent iterative method using low-rank formats satisfying this condition. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank