An application of the Krylov-FSP-SSA method to parameter fitting with maximum likelihood.

Monte Carlo methods such as the stochastic simulation algorithm (SSA) have traditionally been employed in gene regulation problems. However, there has been increasing interest to directly obtain the probability distribution of the molecules involved by solving the chemical master equation (CME). This requires addressing the curse of dimensionality that is inherent in most gene regulation problems. The finite state projection (FSP) seeks to address the challenge and there have been variants that further reduce the size of the projection or that accelerate the resulting matrix exponential. The Krylov-FSP-SSA variant has proved numerically efficient by combining, on one hand, the SSA to adaptively drive the FSP, and on the other hand, adaptive Krylov techniques to evaluate the matrix exponential. Here we apply this Krylov-FSP-SSA to a mutual inhibitory gene network synthetically engineered in Saccharomyces cerevisiae, in which bimodality arises. We show numerically that the approach can efficiently approximate the transient probability distribution, and this has important implications for parameter fitting, where the CME has to be solved for many different parameter sets. The fitting scheme amounts to an optimization problem of finding the parameter set so that the transient probability distributions fit the observations with maximum likelihood. We compare five optimization schemes for this difficult problem, thereby providing further insights into this approach of parameter estimation that is often applied to models in systems biology where there is a need to calibrate free parameters.

Understanding the finite state projection and related methods for solving the chemical master equation.

The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP.

Inference of chromosome selection parameters and missegregation rate in cancer from DNA-sequencing data.

Aneuploidy is frequently observed in cancers and has been linked to poor patient outcome. Analysis of aneuploidy in DNA-sequencing (DNA-seq) data necessitates untangling the effects of the Copy Number Aberration (CNA) occurrence rates and the selection coefficients that act upon the resulting karyotypes. We introduce a parameter inference algorithm that takes advantage of both bulk and single-cell DNA-seq cohorts. The method is based on Approximate Bayesian Computation (ABC) and utilizes CINner, our recently introduced simulation algorithm of chromosomal instability in cancer. We examine three groups of statistics to summarize the data in the ABC routine: (A) Copy Number-based measures, (B) phylogeny tip statistics, and (C) phylogeny balance indices. Using these statistics, our method can recover both the CNA probabilities and selection parameters from ground truth data, and performs well even for data cohorts of relatively small sizes. We find that only statistics in groups A and C are well-suited for identifying CNA probabilities, and only group A carries the signals for estimating selection parameters. Moreover, the low number of CNA events at large scale compared to cell counts in single-cell samples means that statistics in group B cannot be estimated accurately using phylogeny reconstruction algorithms at the chromosome level. As data from both bulk and single-cell DNA-sequencing techniques becomes increasingly available, our inference framework promises to facilitate the analysis of distinct cancer types, differentiation between selection and neutral drift, and prediction of cancer clonal dynamics.

CINner: modeling and simulation of chromosomal instability in cancer at single-cell resolution.

Cancer development is characterized by chromosomal instability, manifesting in frequent occurrences of different genomic alteration mechanisms ranging in extent and impact. Mathematical modeling can help evaluate the role of each mutational process during tumor progression, however existing frameworks can only capture certain aspects of chromosomal instability (CIN). We present CINner, a mathematical framework for modeling genomic diversity and selection during tumor evolution. The main advantage of CINner is its flexibility to incorporate many genomic events that directly impact cellular fitness, from driver gene mutations to copy number alterations (CNAs), including focal amplifications and deletions, missegregations and whole-genome duplication (WGD). We apply CINner to find chromosome-arm selection parameters that drive tumorigenesis in the absence of WGD in chromosomally stable cancer types. We found that the selection parameters predict WGD prevalence among different chromosomally unstable tumors, hinting that the selective advantage of WGD cells hinges on their tolerance for aneuploidy and escape from nullisomy. Direct application of CINner to model the WGD proportion and fraction of genome altered (FGA) further uncovers the increase in CNA probabilities associated with WGD in each cancer type. CINner can also be utilized to study chromosomally stable cancer types, by applying a selection model based on driver gene mutations and focal amplifications or deletions. Finally, we used CINner to analyze the impact of CNA probabilities, chromosome selection parameters, tumor growth dynamics and population size on cancer fitness and heterogeneity. We expect that CINner will provide a powerful modeling tool for the oncology community to quantify the impact of newly uncovered genomic alteration mechanisms on shaping tumor progression and adaptation.

Predicting Time to Relapse in Acute Myeloid Leukemia through Stochastic Modeling of Minimal Residual Disease Based on Clonality Data.

Event-free and overall survival remain poor for patients with acute myeloid leukemia. Chemoresistant clones contributing to relapse arise from minimal residual disease (MRD) or newly-acquired mutations. However, the dynamics of clones comprising MRD is poorly understood. We developed a predictive stochastic model, based on a multitype age-dependent Markov branching process, to describe how random events in MRD contribute to the heterogeneity in treatment response. We employed training and validation sets of patients who underwent whole genome sequencing and for whom mutant clone frequencies at diagnosis and relapse were available. The disease evolution and treatment outcome are subject to stochastic fluctuations. Estimates of malignant clone growth rates, obtained by model fitting, are consistent with published data. Using the estimates from the training set, we developed a function linking MRD and time of relapse, with MRD inferred from the model fits to clone frequencies and other data. An independent validation set confirmed our model. In a third data set, we fitted the model to data at diagnosis and remission and predicted the time to relapse. As a conclusion, given bone marrow genome at diagnosis and MRD at or past remission, the model can predict time to relapse, and help guide treatment decisions to mitigate relapse.

Application of the Moran Model in Estimating Selection Coefficient of Mutated CSF3R Clones in the Evolution of Severe Congenital Neutropenia to Myeloid Neoplasia.

Bone marrow failure (BMF) syndromes, such as severe congenital neutropenia (SCN) are leukemia predisposition syndromes. We focus here on the transition from SCN to pre-leukemic myelodysplastic syndrome (MDS). Stochastic mathematical models have been conceived that attempt to explain the transition of SCN to MDS, in the most parsimonious way, using extensions of standard processes of population genetics and population dynamics, such as the branching and the Moran processes. We previously presented a hypothesis of the SCN to MDS transition, which involves directional selection and recurrent mutation, to explain the distribution of ages at onset of MDS or AML. Based on experimental and clinical data and a model of human hematopoiesis, a range of probable values of the selection coefficient s and mutation rate µ have been determined. These estimates lead to predictions of the age at onset of MDS or AML, which are consistent with the clinical data. In the current paper, based on data extracted from published literature, we seek to provide an independent validation of these estimates. We proceed with two purposes in mind: (i) to determine the ballpark estimates of the selection coefficients and verify their consistency with those previously obtained and (ii) to provide possible insight into the role of recurrent mutations of the G-CSF receptor in the SCN to MDS transition.