David E. Axelrod

Evolution of cooperation among tumor cells

The evolution of cooperation has a well established theoretical framework based on game theory. This approach has made valuable contributions to a wide variety of disciplines, including political science, economics, and evolutionary biology. Existing cancer theory suggests that individual clones of cancer cells evolve independently from one another, acquiring all of the genetic traits or hallmarks necessary to form a malignant tumor. It is also now recognized that tumors are heterotypic, with cancer cells interacting with normal stromal cells within the tissue microenvironment, including endothelial, stromal, and nerve cells. This tumor cell–stromal cell interaction in itself is a form of commensalism, because it has been demonstrated that these nonmalignant cells support and even enable tumor growth. Here, we add to this theory by regarding tumor cells as game players whose interactions help to determine their Darwinian fitness. We marshal evidence that tumor cells overcome certain host defenses by means of diffusible products. Our original contribution is to raise the possibility that two nearby cells can protect each other from a set of host defenses that neither could survive alone. Cooperation can evolve as by-product mutualism among genetically diverse tumor cells. Our hypothesis supplements, but does not supplant, the traditional view of carcinogenesis in which one clonal population of cells develops all of the necessary genetic traits independently to form a tumor. Cooperation through the sharing of diffusible products raises new questions about tumorigenesis and has implications for understanding observed phenomena, designing new experiments, and developing new therapeutic approaches.

Unequal cell division, growth regulation and colony size of mammalian cells: A mathematical model and analysis of experimental data

This work describes mathematically the dynamics of expansion of cell populations from the initial division of single cells to colonies of several hundred cells. This stage of population growth is strongly influenced by stochastic (random) elements including, among others, cell death and quiescence. This results in a wide distribution of colony sizes. Experimental observations of the NIH3T3 cell line as well as for the NIH3T3 cell line transformed with the ras oncogene were obtained for this study. They include the number of cells in 4-day-old colonies initiated from single cells and measurements of sizes of sister cells after division, recorded in the 4-day-old colonies. The sister cell sizes were recorded in a way which enabled investigation of their interdependence. We developed a mathematical model which includes cell growth and unequal cell division, with three possible outcomes of each cell division: continued cell growth and division, quiescence, and cell death. The model is successful in reproducing experimental observations. It provides good fits to colony size distributions for both NIH3T3 mouse fibroblast cells and the same cells transformed with the rasEJ human cancer gene. The difference in colony size distributions could be fitted by assuming similar cell lifetimes (12-13 hr) and similar probabilities of cell death (q = 0.15), but using different probabilities of quiescence, r = 0 for the ras oncogene transformed cells and r = 0.1 for the non-transformed cells. The model also reproduces the evolution of distributions of sizes of cells in colonies, from a single founder cell of any specified size to the stable limit distribution after eight to ten cell divisions. Application of the model explains in what way both random events and deterministic control mechanisms strongly influence cell proliferation at early stages in the expansion of colonies.

Logical analysis of diffuse large B-cell lymphomas

OBJECTIVE The goal of this study is to re-examine the oligonucleotide microarray dataset of Shipp et al., which contains the intensity levels of 6817 genes of 58 patients with diffuse large B-cell lymphoma (DLBCL) and 19 with follicular lymphoma (FL), by means of the combinatorics, optimisation, and logic-based methodology of logical analysis of data (LAD). The motivations for this new analysis included the previously demonstrated capabilities of LAD and its expected potential (1) to identify different informative genes than those discovered by conventional statistical methods, (2) to identify combinations of gene expression levels capable of characterizing different types of lymphoma, and (3) to assemble collections of such combinations that if considered jointly are capable of accurately distinguishing different types of lymphoma. METHODS AND MATERIALS The central concept of LAD is a pattern or combinatorial biomarker, a concept that resembles a rule as used in decision tree methods. LAD is able to exhaustively generate the collection of all those patterns which satisfy certain quality constraints, through a systematic combinatorial process guided by clear optimization criteria. Then, based on a set covering approach, LAD aggregates the collection of patterns into classification models. In addition, LAD is able to use the information provided by large collections of patterns in order to extract subsets of variables, which collectively are able to distinguish between different types of disease. RESULTS For the differential diagnosis of DLBCL versus FL, a model based on eight significant genes is constructed and shown to have a sensitivity of 94.7% and a specificity of 100% on the test set. For the prognosis of good versus poor outcome among the DLBCL patients, a model is constructed on another set consisting also of eight significant genes, and shown to have a sensitivity of 87.5% and a specificity of 90% on the test set. The genes selected by LAD also work well as a basis for other kinds of statistical analysis, indicating their robustness. CONCLUSION These two models exhibit accuracies that compare favorably to those in the original study. In addition, the current study also provides a ranking by importance of the genes in the selected significant subsets as well as a library of dozens of combinatorial biomarkers (i.e. pairs or triplets of genes) that can serve as a source of mathematically generated, statistically significant research hypotheses in need of biological explanation.

A branching process model of gene amplification following chromosome breakage

We have devised a mathematical model of gene amplification utilizing recent experimental observations concerning dihydrofolate reductase (DHFR) gene amplification in CHO cells. The mathematical model, based on a biological model which proposes that acentric elements are the initial intermediates in gene amplification, includes the following features: (1) initiation of amplification by chromosomal breakage to produce an acentric structure; (2) replication of acentric DNA, once per cell cycle; (3) dissociation of replicated acentric DNA; (4) unequal segregation of acentric DNA fragments to daughter cells at mitosis; (5) subsequent reintegration of acentric fragments into chromosomes. These processes are assumed to be independent for each element present in a cell at a given time. Thus, processes of unequal segregation and integration may occur in parallel, not necessarily in a unique sequence, and may be reiterated in one or multiple cell cycles. These events are described mathematically as a Galton-Watson branching process with denumerable infinity of object types. This mathematical model qualitatively and quantitatively reproduces the major elements of the dynamical behavior of DHFR genes observed experimentally. The agreement between the mathematical model and the experimental data lends credence to the biological model proposed by Windle et al. (1991), including the importance of chromosome breakage and subsequent gene deletion resulting from resection of the broken chromosome ends as initial events in gene amplification.

Ductal carcinoma in situ of the breast (DCIS) with heterogeneity of nuclear grade: prognostic effects of quantitative nuclear assessment

BackgroundPreviously, 50% of patients with breast ductal carcinoma in situ (DCIS) had more than one nuclear grade, and neither worst nor predominant nuclear grade was significantly associated with development of invasive carcinoma. Here, we used image analysis in addition to histologic evaluation to determine if quantification of nuclear features could provide additional prognostic information and hence impact prognostic assessments.MethodsNuclear image features were extracted from about 200 nuclei of each of 80 patients with DCIS who underwent lumpectomy alone, and received no adjuvant systemic therapy. Nuclear images were obtained from 20 representative nuclei per duct, from each of a group of 5 ducts, in two separate fields, for 10 ducts. Reproducibility of image analysis features was determined, as was the ability of features to discriminate between nuclear grades. Patient information was available about clinical factors (age and method of DCIS detection), pathologic factors (DCIS size, nuclear grade, margin size, and amount of parenchymal involvement), and 39 image features (morphology, densitometry, and texture). The prognostic effects of these factors and features on the development of invasive breast cancer were examined with Cox step-wise multivariate regression.ResultsDuplicate measurements were similar for 89.7% to 97.4% of assessed image features. For the pooled assessment with ~200 nuclei per patient, a discriminant function with one densitometric and two texture features was significantly (p < 0.001) associated with nuclear grading, and provided 78.8% correct jackknifed classification of a patients nuclear grade. In multivariate assessments, image analysis nuclear features had significant prognostic associations (p ≤ 0.05) with the development of invasive breast cancer. Texture (difference entropy, p < 0.001; contrast, p < 0.001; peak transition probability, p = 0.01), densitometry (range density, p = 0.004), and measured margin (p = 0.05) were associated with development of invasive disease for the pooled data across all ducts.ConclusionImage analysis provided reproducible assessments of nuclear features which quantitated differences in nuclear grading for patients. Quantitative nuclear image features indicated prognostically significant differences in DCIS, and may contribute additional information to prognostic assessments of which patients are likely to develop invasive disease.

A calibrated agent-based computer model of stochastic cell dynamics in normal human colon crypts useful for in silico experiments.

BackgroundNormal colon crypts consist of stem cells, proliferating cells, and differentiated cells. Abnormal rates of proliferation and differentiation can initiate colon cancer. We have measured the variation in the number of each of these cell types in multiple crypts in normal human biopsy specimens. This has provided the opportunity to produce a calibrated computational model that simulates cell dynamics in normal human crypts, and by changing model parameter values, to simulate the initiation and treatment of colon cancer.ResultsAn agent-based model of stochastic cell dynamics in human colon crypts was developed in the multi-platform open-source application NetLogo. It was assumed that each cell’s probability of proliferation and probability of death is determined by its position in two gradients along the crypt axis, a divide gradient and in a die gradient. A cell’s type is not intrinsic, but rather is determined by its position in the divide gradient. Cell types are dynamic, plastic, and inter-convertible. Parameter values were determined for the shape of each of the gradients, and for a cell’s response to the gradients. This was done by parameter sweeps that indicated the values that reproduced the measured number and variation of each cell type, and produced quasi-stationary stochastic dynamics. The behavior of the model was verified by its ability to reproduce the experimentally observed monocolonal conversion by neutral drift, the formation of adenomas resulting from mutations either at the top or bottom of the crypt, and by the robust ability of crypts to recover from perturbation by cytotoxic agents. One use of the virtual crypt model was demonstrated by evaluating different cancer chemotherapy and radiation scheduling protocols.ConclusionsA virtual crypt has been developed that simulates the quasi-stationary stochastic cell dynamics of normal human colon crypts. It is unique in that it has been calibrated with measurements of human biopsy specimens, and it can simulate the variation of cell types in addition to the average number of each cell type. The utility of the model was demonstrated with in silico experiments that evaluated cancer therapy protocols. The model is available for others to conduct additional experiments.

Fluctuation test for two-stage mutations: application to gene amplification

The determination of mutation rates is an important experimental procedure for characterizing mutation processes. The accepted method of determining mutation rates, the fluctuation test, was introduced by Luria and Delbrück in 1943. Since then it has been applied to various microorganisms and cells. The Luria-Delbrück test is based on a restrictive hypothesis of mutations being due to single irreversible events. However, some inherited changes in phenotype, like gene amplification, may be due to two or more genetic changes, some of which may be reversible. The Luria-Delbrück model of mutation was compared to other models which included reversibility and more than one mutation stage. The Luria-Delbrück model has been confirmed to be consistent with the original bacteriophage resistance data. However, for gene amplification this model gives incompatible estimates of mutation rates by the P0 and r methods. Relaxing the hypotheses of the single-stage models did not improve the fit. In contrast, a two-stage reversible model provided a fit. Analysis of gene amplification data by the two-stage reversible model provides new information, including estimates of rates for each of the two forward stages and of the reverse step.

Assessing genetic markers of tumour progression in the context of intratumour heterogeneity.

This is a report from the Kananaskis working group on quantitative methods in tumour heterogeneity. Tumour progression is currently believed to result from genetic instability and consequent acquisition of new genetic properties in some of the tumour cells. Cross-sectional assessment of genetic markers for human tumours requires quantifiable measures of intratumour heterogeneity for each parameter or characteristic observed; the relevance of heterogeneity to tumour progression can best be ascertained by repeated assessment along a tumour progressional time line. This paper outlines experimental and analytic considerations that, with repeated use, should lead to a better understanding of tumour heterogeneity, and hence, to improvements in patient diagnosis and therapy. Four general principles were agreed upon at the Symposium: (1) the concept of heterogeneity requires a quantifiable definition so that it can be assessed repeatably; (2) the quantification of heterogeneity is necessary so that testable hypotheses may be formulated and checked to determine the degree of support from observed data; (3) it is necessary to consider (a) what is being measured, (b) what is currently measurable, and (c) what should be measured; and (4) the proposal of working models is a useful step that will assist our understanding of the origins and significance of heterogeneity in tumours. The properties of these models should then be studied so that hypotheses may be refined and validated.

Effect of Quantitative Nuclear Image Features on Recurrence of Ductal Carcinoma In Situ (DCIS) of the Breast

Background Nuclear grade has been associated with breast DCIS recurrence and progression to invasive carcinoma; however, our previous study of a cohort of patients with breast DCIS did not find such an association with outcome. Fifty percent of patients had heterogeneous DCIS with more than one nuclear grade. The aim of the current study was to investigate the effect of quantitative nuclear features assessed with digital image analysis on ipsilateral DCIS recurrence. Methods Hematoxylin and eosin stained slides for a cohort of 80 patients with primary breast DCIS were reviewed and two fields with representative grade (or grades) were identified by a Pathologist and simultaneously used for acquisition of digital images for each field. Van Nuys worst nuclear grade was assigned, as was predominant grade, and heterogeneous grading when present. Patients were grouped by heterogeneity of their nuclear grade: Group A: nuclear grade 1 only, nuclear grades 1 and 2, or nuclear grade 2 only (32 patients), Group B: nuclear grades 1, 2 and 3, or nuclear grades 2 and 3 (31 patients), Group 3: nuclear grade 3 only (17 patients). Nuclear fine structure was assessed by software which captured thirty-nine nuclear feature values describing nuclear morphometry, densitometry, and texture. Step-wise forward Cox regressions were performed with previous clinical and pathologic factors, and the new image analysis features. Results Duplicate measurements were similar for 89.7% to 97.4% of assessed image features. The rate of correct classification of nuclear grading with digital image analysis features was similar in the two fields, and pooled assessment across both fields. In the pooled assessment, a discriminant function with one nuclear morphometric and one texture feature was significantly (p = 0.001) associated with nuclear grading, and provided correct jackknifed classification of a patients nuclear grade for Group A (78.1%), Group B (48.4%), and Group C (70.6%). The factors significantly associated with DCIS recurrence were those previously found, type of initial presentation (p = 0.03) and amount of parenchymal involvement (p = 0.05), along with the morphometry image feature of ellipticity (p = 0.04). Conclusion Analysis of nuclear features measured by image cytometry may contribute to the classification and prognosis of breast DCIS patients with more than one nuclear grade.

The importance of clonal heterogeniety and interexperiment variability in modeling the eukaryotic cell cycle

Abstract Many models of the eukaryotic cell cycle find their motivation and/or justification in the analysis of observed interdivision time. In the presence of clonal heterogeneity, however, certain measures such as correlations are skewed in the positive direction. We show how biases arise from pooling heterogeneous data and, with our own data, show how to test for it and correct for its effects in parameter estimates. We discuss what implications this has for models of the eukaryotic cell cycle.