Multi-resolution modeling of a discrete stochastic process identifies causes of cancer

1 INTRODUCTION . Numerous domains involve modeling highly non-stationary discrete-time and integer-valued stochastic processes where event counts vary dramatically over time or space . An important open problem of this nature in biology is understanding the stochastic process by which mutations arise across the genome . This is central to identifying mutations that drive cancer emergence ( Lawrence et al. , 2013 ) . Tumor drivers provide a cellular growth advantage to cells by altering the function of a genomic element such as a gene or regulatory feature ( e.g . promoter ) . Drivers are identifiable because they reoccur across tumors , but there are two major challenges to detecting such recurrence . First , driver mutations are rare and their signal is hidden by the thousands of passenger mutations that passively and stochastically accumulate in tumors ( Stratton et al. , 2009 ; Martincorena & Campbell , 2015 ) . Second , because functional elements vary dramatically in size ( genes : 103-106 bases ; regulatory elements : 101-103 bases ; and single positions ) , driver mutations accumulate across regions that vary many orders of magnitude . Accurately predicting the stochastic accumulation of passenger mutations at multiple scales is necessary to reveal the subtle recurrence of driver mutations across the genome . Here , we introduce the split-Poisson Gamma ( SPG ) process , an extension of the Poisson-Gamma distribution , to efficiently model a non-stationary discrete stochastic process at numerous length scales . The model first approximates quasi-stationary regional rate parameters within small windows ; it then projects these estimates to arbitrary regions in linear time ( 10-15 minutes for genome-wide inference ) . This approach is in contrast to existing efforts that model fixed regions and require computationally expensive retraining ( e.g . over 5 hours ) to predict over multiple scales of interest ( Nik-Zainal et al. , 2016 ; Martincorena et al. , 2017 ) . We apply our framework to model cancer-specific mutation patterns ( fig . 1 ) . We perform data-driven training of our model ’ s parameters and show that it more accurately captures mutation patterns than existing methods on simulated and real data . We demonstrate the power of our multi-resolution approach by identifying drivers across functional ∗Authors contributed equally to this work . elements : genes , regulatory features , and single base mutations . Despite the method having no knowledge of genome structure , it detects nearly all gene drivers present in over 5 % of samples while making no false discoveries and detects all previously characterized regulatory drivers . Detected events also include novel candidate drivers , providing promising targets for future investigation . 1.1 PREVIOUS WORK . Numerous methods exist for modeling stationary stochastic processes ( Lindsey , 2004 ) . Far fewer exist for non-stationary processes because they are difficult to capture with the covariance functions of parametric models ( Risser , 2016 ) . Non-stationary kernels have been introduced for Gaussian processes ( Paciorek & Schervish , 2004 ) , but these may not be tractable on large datasets due to their computational complexity . More recently , there has been work developing Poisson-gamma models for dynamical systems ( Schein et al. , 2016 ; Guo et al. , 2018 ) , but these methods have focused on learning relationships between count variables , not predicting counts based on continuous covariates . In the particular case of modeling mutation patterns across the cancer genome , numerous computational methods exist to model mutation rates within well-understood genomic contexts such as genes ( Lawrence et al. , 2013 ; Martincorena et al. , 2017 ; Wadi et al. , 2017 ; Mularoni et al. , 2016 ; Juul et al. ) . These models account for < 4 % of the genome ( Rheinbay et al. , 2020 ) . They are not applicable in non-coding regions , where the majority of mutations occur ( Gloss & Dinger , 2018 ) . A handful of methods to model genome-wide mutation rates have been introduced ( Polak et al. , 2015 ; Nik-Zainal et al. , 2016 ; Bertl et al. , 2018 ) . However , they operate on a single length-scale or set of regions and require computationally expensive retraining to predict over each new length-scale . Several methods rely on Poisson or binomial regression ; however , previous work has extensively documented that mutation counts data are over-dispersed , leading these models to underestimate variance and yield numerous false-positive driver predictions ( Lochovsky et al. , 2015 ; Martincorena et al. , 2017 ; Juul et al. , 2019 ) . Negative binomial regression has recently been used to account for over-dispersion ( Nik-Zainal et al. , 2016 ) and perform genome-wide mutation modeling and driver detection . However , resolution was coarse , and it only found a few , highly recurrent driver mutations . 1.2 OUR CONTRIBUTIONS . This work makes three key contributions : 1 ) we introduce an extension of the Poisson-Gamma distribution to model non-stationary discrete stochastic processes at any arbitrary length scale without retraining ; 2 ) we apply the framework to capture cancer-specific mutation rates with unprecedented accuracy , resolution , and efficiency ; and 3 ) we perform a multi-scale search for cancer driver mutations genome-wide , including the first-ever base-resolution scan of the whole genome . This search yields several new candidate driver events in the largely unexplored non-coding genome , which we are working on validating with experimental collaborators . Crucially , our approach allows fast , efficient , and accurate searches for driver elements and mutations anywhere in the genome without requiring arduous retraining of a model , a feat which is not possible with existing approaches . 2 MULTI-RESOLUTION MODELING OF A NON-STATIONARY DISCRETE STOCHASTIC PROCESS . We consider a non-stationary discrete stochastic process { Mi ; i = 1,2 , ... } where Mi is the integervalued event count at position i . Associated with each position i is a real-valued , L-dimensional feature vector ηi that determines the instantaneous event rate λi via an unknown function . Thus a region R = { i , i+1 , ... , i+N } of N contiguous positions is characterized by an L×N feature matrix ηR and an event count XR = ∑i∈R Mi . As training data , ηR , XR , and Mi are observed for some set of regions { R ∈T } . Then given a set of feature matrices from unobserved regions { ηR ; R ∈H } , the challenge is to predict the distribution of event counts over any arbitrary set I of unseen positions that may or may not be contiguous . Real-world examples include traders in a stock market , packets delivered to routers in a network , and mutations accumulating at positions in the genome . 2.1 THE SPLIT-POISSON-GAMMA PROCESS . We assume that the process is near-stationary within a small enough region R= { i , i+1 , ... , i+N } and that the L×N covariate matrix ηR is observed . Thus the rate of events λR within R is approximately constant and associated with ηR , albeit in an unknown way . A number of events ( XR ) may occur within R dependent on λR and are then stochastically distributed to individual positions within R , implying a hierarchical factorization of the scalar random variables λR , XR , and Mi ( fig . 1e ) as Pr ( Mi = k , XR , λR ; ηR ) = Pr ( Mi = k|XR ; ηR ) Pr ( XR|λR ; ηR ) Pr ( λR ; ηR ) . ( 1 ) XR and λR are unknown nuisance variables and are marginalized in general as Pr ( Mi = k|ηR ) = ∫ ∞ 0 Pr ( λR ; ηR ) ∞ ∑ XR=k Pr ( Mi = k|XR ; ηR ) Pr ( XR|λR ; ηR ) dλR . ( 2 ) Since applications often require many posterior predictions over regions of varying sizes , we propose a prior parameterization that builds on the success and flexibility of the classical Poisson-Gamma distribution while ensuring the marginalization has an easy-to-compute posterior distribution : λR ∼ Gamma ( αR , θR ) ( 3 ) XR ∼ Poisson ( λR ) ( 4 ) Mi ∼ Binomial ( XR , p̃i ) ( 5 ) where αR and θR are shape and scale parameters dependent on ηR , pi is the time-averaged probability of an event at i and p̃i = pi ∑ j∈R p j , the normalized probability within R. A plate diagram of the hierarchical model is presented in fig . 1e . The above formulation provides a simple , closed form solution to eq . ( 2 ) as a negative binomial ( NB ) distribution ( See Appendix for details ) : Pr ( Mi = k|αR , θR , p̃i ; ηr ) = NB ( k ; αR , 1 1+θR · p̃i ) . ( 6 ) Eq . 5 implicitly assumes that events are distributed independently to units within R. Exploiting this assumption , eq . ( 6 ) immediately generalizes to consider any set of units I ⊆ R as Pr ( ∑ i∈I Mi = k|αR , θR , { p̃i } i∈I ; ηR ) = NB ( k ; αR , 1 1+θR ·∑i∈I p̃i ) . ( 7 ) The above formulation is an extension of the classical Poisson-Gamma distribution whereby the Poisson is randomly split by a binomial . We term this a split-Poisson-Gamma ( SPG ) process . While the derivation of the SPG solution makes simplifying assumptions , the benefit is that the parameters αR and θR need to be estimated only once for each non-overlapping region R. Estimates for a region of any other size can then be computed in constant time from eq . ( 7 ) . If a new region R′ is larger than R , we approximate the gamma distribution in a super-region containing R′ as a superposition of the previously inferred parameters of each region of size R within the super-region ( see section 2.2 ) . 2.2 INFERRING REGIONAL RATE PARAMETERS . The statistical power of SPG depends on accurate estimation of the regional gamma rate parameters αR and θR . We propose a variational approach to enable flexible , accurate , and non-linear inference of these parameters from a set of covariates . Let G ( α , θ ) be a gamma distribution . By the central limit theorem , limα→∞ G ( α , θ ) = N ( µ , σ2 ) where µ = αθ and σ2 = αθ 2 . We thus use a Gaussian process ( GP ) to non-linearly map covariates to regional estimates for µR and σ2R . The variational estimates for the gamma parameters are then αR = µ2R/σ 2 R , θR = µR/σ 2 R ( 8 ) For a super-region R′ = Ri +R j , µR′ = µRi +µR j and σ2R′ = σ 2 Ri +σ 2 R j . A limitation of this approach is that GPs can only operate on vectors of covariates . Thus a dimensionality reduction method must be applied to the input matrix ηR . In cases where ηR includes spatial relationships , a convolutional neural network can be a powerful approach to dimension-reduction ; however , other approaches are feasible ( see section 3.2 and section 5.1 ) . 2.3 INFERRING TIME-AVERAGED EVENT PROBABILITIES . The time-averaged parameters { pi ; i = 1,2 , ... } must also be inferred . Crucially , as seen in eq . ( 5 ) , these parameters are never used directly ; instead , they are always renormalized to sum to one within a region of interest . Thus , estimates do not need to reflect the absolute probability of an event at i but merely the relative rate of events between positions . Indeed , because of the renormalization procedure , the estimates need not even be a true probability distribution . Estimating pi can thus be accomplished by clustering units with similar relative rates of events . How this clustering should be performed will depend on the application of interest ( see section 3.3 for a concrete example ) .

Summary: The authors present the split Poisson Gamma (SPG) distribution, an extension of the Poisson-Gamma distribution, to model a discrete non-stationary stochastic process. SPG has an analytical posterior allowing accurate prediction after the model parameters have been inferred a single time. The authors apply the SPG to model tumor mutation rates and show that model parameters can be accurately inferred from high-dimensional epigenetic data. This is achieved through a combination of CNNs, GPs and MLE. The results are promising in detecting tumor drivers such as genes, regulatory structures and base-pairs.