Somatic mosaicism and cancer: inference based on a conditional Luria-Delbrück distribution

Reuniting philosophy and science to advance cancer research

Cancers rely on multiple, heterogeneous processes at different scales, pertaining to many biomedical fields. Therefore, understanding cancer is necessarily an interdisciplinary task that requires placing specialised experimental and clinical research into a broader conceptual, theoretical, and methodological framework. Without such a framework, oncology will collect piecemeal results, with scant dialogue between the different scientific communities studying cancer. We argue that one important way forward in service of a more successful dialogue is through greater integration of applied sciences (experimental and clinical) with conceptual and theoretical approaches, informed by philosophical methods. By way of illustration, we explore six central themes: (i) the role of mutations in cancer; (ii) the clonal evolution of cancer cells; (iii) the relationship between cancer and multicellularity; (iv) the tumour microenvironment; (v) the immune system; and (vi) stem cells. In each case, we examine open questions in the scientific literature through a philosophical methodology and show the benefit of such a synergy for the scientific and medical understanding of cancer.

The number of neutral mutants in an expanding Luria-Delbrück population is approximately Fréchet

Background: A growing population of cells accumulates mutations. A single mutation early in the growth process carries forward to all descendant cells, causing the final population to have a lot of mutant cells. When the first mutation happens later in growth, the final population typically has fewer mutants. The number of mutant cells in the final population follows the Luria-Delbrück distribution. The mathematical form of the distribution is known only from its probability generating function. For larger populations of cells, one typically uses computer simulations to estimate the distribution. Methods: This article searches for a simple approximation of the Luria-Delbrück distribution, with an explicit mathematical form that can be used easily in calculations. Results: The Fréchet distribution provides a good approximation for the Luria-Delbrück distribution for neutral mutations, which do not cause a growth rate change relative to the original cells. Conclusions: The Fréchet distribution apparently provides a good match through its description of extreme value problems for multiplicative processes such as exponential growth.

Control of tumor growth distributions through kinetic methods

The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker-Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.

Single-Cell Transcriptomics Analysis Identifies Nuclear Protein 1 as a Regulator of Docetaxel Resistance in Prostate Cancer Cells

The majority of patients with prostate cancer treated with docetaxel develop resistance to it. To better understand the mechanism behind the acquisition of resistance, we conducted single-cell RNA-sequencing (scRNA-seq) of docetaxel-sensitive and -resistant variants of DU145 and PC3 prostate cancer cell lines. Overall, sensitive and resistant cells clustered separately. Differential gene expression analysis between resistant and sensitive cells revealed 182 differentially expressed genes common to both prostate cancer cell lines. A subset of these genes gave a gene expression profile in the resistant transcriptome-like-sensitive cells similar to the resistant cells. Exploration for functional gene pathways identified 218 common pathways between the two cell lines. Protein ubiquitination was the most differentially regulated pathway and was enriched in the resistant cells. Transcriptional regulator analysis identified 321 potential regulators across both cell lines. One of the top regulators identified was nuclear protein 1 (NUPR1). In contrast to the single-cell analysis, bulk analysis of the cells did not reveal NUPR1 as a promising candidate. Knockdown and overexpression of NUPR1 in the prostate cancer cells demonstrated that NUPR1 confers docetaxel resistance in both cell lines. Collectively, these data demonstrate the utility of scRNA-seq to identify regulators of drug resistance. Furthermore, NUPR1 was identified as a mediator of prostate cancer drug resistance, which provides the rationale to explore NUPR1 and its target genes for reversal of docetaxel resistance. IMPLICATIONS: Using single-cell sequencing of prostate cancer, we show that NUPR1 plays a role in docetaxel resistance.

Somatic mosaicism and disease

The large number of cell divisions required to make a human body inevitably leads to the accumulation of somatic mutations. Such mutations cause individuals to be somatic mosaics. Recent advances in genomic technology now allow measurement of somatic diversity. Initial studies confirmed the expected high levels of somatic mutations within individuals. Going forward, the big questions concern the degree to which those somatic mutations influence disease. Theory predicts that the frequency of mutant cells should vary greatly between individuals. Such somatic mutational variability between individuals could explain much of the diversity in the risk of disease. But how variable is mosaicism between individuals in reality? What is the relation between the fraction of cells carrying a predisposing mutation and the risk of disease? What kinds of heritable somatic change lead to disease besides classical DNA mutations? What molecular processes connect a predisposing somatic change to disease? We know that predisposing somatic mutations strongly influence the onset of cancer. Likewise, neurodegenerative diseases may often begin from somatically mutated cells. If so, both neurodegeneration and cancer may be diseases of later life for which much of the risk may be set by early life somatic mutations.

Ultra-fast local-haplotype variant calling using paired-end DNA-sequencing data reveals somatic mosaicism in tumor and normal blood samples

Somatic mosaicism refers to the existence of somatic mutations in a fraction of somatic cells in a single biological sample. Its importance has mainly been discussed in theory although experimental work has started to emerge linking somatic mosaicism to disease diagnosis. Through novel statistical modeling of paired-end DNA-sequencing data using blood-derived DNA from healthy donors as well as DNA from tumor samples, we present an ultra-fast computational pipeline, LocHap that searches for multiple single nucleotide variants (SNVs) that are scaffolded by the same reads. We refer to scaffolded SNVs as local haplotypes (LH). When an LH exhibits more than two genotypes, we call it a local haplotype variant (LHV). The presence of LHVs is considered evidence of somatic mosaicism because a genetically homogeneous cell population will not harbor LHVs. Applying LocHap to whole-genome and whole-exome sequence data in DNA from normal blood and tumor samples, we find wide-spread LHVs across the genome. Importantly, we find more LHVs in tumor samples than in normal samples, and more in older adults than in younger ones. We confirm the existence of LHVs and somatic mosaicism by validation studies in normal blood samples. LocHap is publicly available at http://www.compgenome.org/lochap.

The mathematics of cancer: integrating quantitative models

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

The Impact of Microenvironmental Heterogeneity on the Evolution of Drug Resistance in Cancer Cells

Therapeutic resistance arises as a result of evolutionary processes driven by dynamic feedback between a heterogeneous cell population and environmental selective pressures. Previous studies have suggested that mutations conferring resistance to epidermal growth factor receptor (EGFR) tyrosine kinase inhibitors (TKI) in non-small-cell lung cancer (NSCLC) cells lower the fitness of resistant cells relative to drug-sensitive cells in a drug-free environment. Here, we hypothesize that the local tumor microenvironment could influence the magnitude and directionality of the selective effect, both in the presence and absence of a drug. Using a combined experimental and computational approach, we developed a mathematical model of preexisting drug resistance describing multiple cellular compartments, each representing a specific tumor environmental niche. This model was parameterized using a novel experimental dataset derived from the HCC827 erlotinib-sensitive and -resistant NSCLC cell lines. We found that, in contrast to in the drug-free environment, resistant cells may hold a fitness advantage compared to parental cells in microenvironments deficient in oxygen and nutrients. We then utilized the model to predict the impact of drug and nutrient gradients on tumor composition and recurrence times, demonstrating that these endpoints are strongly dependent on the microenvironment. Our interdisciplinary approach provides a model system to quantitatively investigate the impact of microenvironmental effects on the evolutionary dynamics of tumor cells.

Evolution of acquired resistance to anti-cancer therapy

Acquired drug resistance is a major limitation for the successful treatment of cancer. Resistance can emerge due to a variety of reasons including host environmental factors as well as genetic or epigenetic alterations in the cancer cells. Evolutionary theory has contributed to the understanding of the dynamics of resistance mutations in a cancer cell population, the risk of resistance pre-existing before the initiation of therapy, the composition of drug cocktails necessary to prevent the emergence of resistance, and optimum drug administration schedules for patient populations at risk of evolving acquired resistance. Here we review recent advances towards elucidating the evolutionary dynamics of acquired drug resistance and outline how evolutionary thinking can contribute to outstanding questions in the field.

Mutator/Hypermutable fetal/juvenile metakaryotic stem cells and human colorectal carcinogenesis

Adult age-specific colorectal cancer incidence rates increase exponentially from maturity, reach a maximum, then decline in extreme old age. Armitage and Doll (1) postulated that the exponential increase resulted from "n" mutations occurring throughout adult life in normal "cells at risk" that initiated the growth of a preneoplastic colony in which subsequent "m" mutations promoted one of the preneoplastic "cells at risk" to form a lethal neoplasia. We have reported cytologic evidence that these "cells at risk" are fetal/juvenile organogenic, then preneoplastic metakaryotic stem cells. Metakaryotic cells display stem-like behaviors of both symmetric and asymmetric nuclear divisions and peculiarities such as bell shaped nuclei and amitotic nuclear fission that distinguish them from embryonic, eukaryotic stem cells. Analyses of mutant colony sizes and numbers in adult lung epithelia supported the inferences that the metakaryotic organogenic stem cells are constitutively mutator/hypermutable and that their contributions to cancer initiation are limited to the fetal/juvenile period. We have amended the two-stage model of Armitage and Doll and incorporated these several inferences in a computer program CancerFit v.5.0. We compared the expectations of the amended model to adult (15-104 years) age-specific colon cancer rates for European-American males born 1890-99 and observed remarkable concordance. When estimates of normal colonic fetal/juvenile APC and OAT gene mutation rates (∼2-5 × 10(-5) per stem cell doubling) and preneoplastic colonic gene loss rates (∼8 × 10(-3)) were applied, the model was in accordance only for the values of n = 2 and m = 4 or 5.

Mean field mutation dynamics and the continuous Luria-Delbrück distribution

The Luria-Delbrück mutation model has a long history and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using some mathematical tools from nonlinear statistical physics. Starting from the classical formulations we derive the corresponding differential models and show that under a suitable mean field scaling they correspond to generalized Fokker-Planck equations for the mutants distribution whose solutions are given by the corresponding Luria-Delbrück distribution. Numerical results confirming the theoretical analysis are also presented.

Fitness conferred by BCR-ABL kinase domain mutations determines the risk of pre-existing resistance in chronic myeloid leukemia

Chronic myeloid leukemia (CML) is the first human malignancy to be successfully treated with a small molecule inhibitor, imatinib, targeting a mutant oncoprotein (BCR-ABL). Despite its successes, acquired resistance to imatinib leads to reduced drug efficacy and frequent progression of disease. Understanding the characteristics of pre-existing resistant cells is important for evaluating the benefits of first-line combination therapy with second generation inhibitors. However, due to limitations of assay sensitivity, determining the existence and characteristics of resistant cell clones at the start of therapy is difficult. Here we combined a mathematical modeling approach using branching processes with experimental data on the fitness changes (i.e., changes in net reproductive rate) conferred by BCR-ABL kinase domain mutations to investigate the likelihood, composition, and diversity of pre-existing resistance. Furthermore, we studied the impact of these factors on the response to tyrosine kinase inhibitors. Our approach predicts that in most patients, there is at most one resistant clone present at the time of diagnosis of their disease. Interestingly, patients are no more likely to harbor the most aggressive, pan-resistant T315I mutation than any other resistance mutation; however, T315I cells on average establish larger-sized clones at the time of diagnosis. We established that for patients diagnosed late, the relative benefit of combination therapy over monotherapy with imatinib is significant, while this benefit is modest for patients with a typically early diagnosis time. These findings, after pre-clinical validation, will have implications for the clinical management of CML: we recommend that patients with advanced-phase disease be treated with combination therapy with at least two tyrosine kinase inhibitors.

Seeing beyond the average cell: branching process models of cell proliferation, differentiation, and death during mouse brain development

We develop a family of branching process models to study cerebral cortical development at the level of individual neural stem and progenitor cells (NS/PCs) and the neurons they produce. Population-level data about "the average NS/PC" is incorporated as constraints for exploring (i) heterogeneity in the proliferative neural cell types and (ii) variability in daughter cell fate decision making. Preliminary studies demonstrate this variability, generate testable hypotheses about heterogeneity, and motivate new experiments moving forward.

Evolution in health and medicine Sackler colloquium: Somatic evolutionary genomics: mutations during development cause highly variable genetic mosaicism with risk of cancer and neurodegeneration

Somatic mutations must happen often during development because of the large number of cell divisions to expand from a single-cell zygote to a full organism. A mutation in development carries forward to all descendant cells, causing genetic mosaicism. Widespread genetic mosaicism may influence diseases that derive from a few genetically altered cells, such as cancer. I show how to predict the expected amount of mosaicism and the variation in mosaicism between individuals. I then calculate the predicted risk of cancer derived from developmental mutations. The calculations show that a significant fraction of cancer in later life likely arises from developmental mutations in early life. In addition, much of the variation in the risk of cancer between individuals may arise from variation in the degree of genetic mosaicism set in early life. I also suggest that certain types of neurodegeneration, such as amyotrophic lateral sclerosis (ALS), may derive from a small focus of genetically altered cells. If so, then the risk of ALS would be influenced by developmental mutations and the consequent variation in genetic mosaicism. New technologies promise the ability to measure genetic mosaicism by sampling a large number of cellular genomes within an individual. The sampling of many genomes within an individual will eventually allow one to reconstruct the cell lineage history of genetic change in a single body. Somatic evolutionary genomics will follow from this technology, providing new insight into the origin and progression of disease with increasing age.

Evolutionary theory of cancer

As Theodosius Dobzhansky famously noted in 1973, "Nothing in biology makes sense except in the light of evolution," and cancer is no exception to this rule. Our understanding of cancer initiation, progression, treatment, and resistance has advanced considerably by regarding cancer as the product of evolutionary processes. Here we review the literature of mathematical models of cancer evolution and provide a synthesis and discussion of the field.

Clonal diversity in carcinomas: its implications for tumour progression and the contribution made to it by epithelial-mesenchymal transitions

The progression of tumours to malignancy is commonly considered to arise through lineal evolution, a process in which mutations conferring pro-oncogenic cellular phenotypes are acquired by a succession of ever-more dominant clones. However, this model is at odds with the persistent polyclonality observed in many cancers. We propose that an alternative mechanism for tumour progression, called interclonal cooperativity, is likely to play a role at stages of tumour progression when mutations cause microenvironmental changes, such as occur with epithelial-mesenchymal transitions (EMTs). Interclonal cooperativity occurs when cancer cell-cancer cell interactions produce an emergent malignant phenotype from individually non-malignant clones. In interclonal cooperativity, the oncogenic mutations occur in different clones within the tumour that complement each other and cooperate in order to drive progression. This reconciles the accepted genetic and evolutionary basis of cancers with the observed polyclonality in tumours. Here, we provide a conceptual basis for examining the importance of cancer cell-cancer cell interactions to the behaviour of tumours and propose specific mechanisms by which clonal diversity in tumours, including that provided by EMTs, can drive the progression of tumours to malignancy.

The evolution of two mutations during clonal expansion

Knudson's two-hit hypothesis proposes that two genetic changes in the RB1 gene are the rate-limiting steps of retinoblastoma. In the inherited form of this childhood eye cancer, only one mutation emerges during somatic cell divisions while in sporadic cases, both alleles of RB1 are inactivated in the growing retina. Sporadic retinoblastoma serves as an example of a situation in which two mutations are accumulated during clonal expansion of a cell population. Other examples include evolution of resistance against anticancer combination therapy and inactivation of both alleles of a metastasis-suppressor gene during tumor growth. In this article, we consider an exponentially growing population of cells that must evolve two mutations to (i) evade treatment, (ii) make a step toward (invasive) cancer, or (iii) display a disease phenotype. We calculate the probability that the population has evolved both mutations before it reaches a certain size. This probability depends on the rates at which the two mutations arise; the growth and death rates of cells carrying none, one, or both mutations; and the size the cell population reaches. Further, we develop a formula for the expected number of cells carrying both mutations when the final population size is reached. Our theory establishes an understanding of the dynamics of two mutations during clonal expansion.

Can loss of apoptosis protect against cancer?

Cells of higher organisms can commit suicide in response to genomic alterations, a process called programmed cell death. Although it is commonly thought that the loss of programmed cell death is required for carcinogenesis, we argue that the situation is more complex and that the loss of programmed cell death can have the converse effect, preventing cancer progression. If the death rate of cancer cells is low, fewer cell divisions are required for the tumor to reach a certain size, resulting in the presence of fewer mutant cells. Therefore, the chances of overcoming potential selective barriers are reduced, rendering the failure of pathogenic progression probable. However, if there is a higher cell death rate, more cell divisions need to occur for the tumor to reach a certain size, resulting in the presence of more mutant cells and in an increased probability of overcoming selective barriers and cancer progression.

Evolution of resistance during clonal expansion

Acquired drug resistance is a major limitation for cancer therapy. Often, one genetic alteration suffices to confer resistance to an otherwise successful therapy. However, little is known about the dynamics of the emergence of resistant tumor cells. In this article, we consider an exponentially growing population starting from one cancer cell that is sensitive to therapy. Sensitive cancer cells can mutate into resistant ones, which have relative fitness alpha prior to therapy. In the special case of no cell death, our model converges to the one investigated by Luria and Delbrück. We calculate the probability of resistance and the mean number of resistant cells once the cancer has reached detection size M. The probability of resistance is an increasing function of the detection size M times the mutation rate u. If Mu << 1, then the expected number of resistant cells in cancers with resistance is independent of the mutation rate u and increases with M in proportion to M(1-1/alpha) for advantageous mutants with relative fitness alpha>1, to l nM for neutral mutants (alpha = 1), but converges to an upper limit for deleterious mutants (alpha<1). Further, the probability of resistance and the average number of resistant cells increase with the number of cell divisions in the history of the tumor. Hence a tumor subject to high rates of apoptosis will show a higher incidence of resistance than expected on its detection size only.