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Targeting the adaptability of heterogeneous aneuploids – Geometrical model of adaptation
The mathematical modelling of the relationship between the robustness and evolvability of heterogeneous cell populations started as an observation that in the aneuploid growth assay data from (Pavelka et al. 2010), that the relative fitness of aneuploids compared to euploids followed a peculiar relationship. Aneuploids always performed worse off that euploid cells when grown in rich medium. As the intensity of stress coming from the environment increased, the mean fitness of aneuploids relative to euploids dropped. However, at the mean relative fitness of aneuploids dropped, the standard deviation of the relative fitness also increased in a way that guaranteed that there always were aneuploid outliers with a fitness superior to the euploids. The degree to which those outlier aneuploids fared better than euploids also increased with the intensity of the stress. Due to the design of the assay that was measuring the fitness and post-screening validation steps, we were sure that this heterogeneity was not due to the measurement noise at low fitness. Instead, this pattern was indicating that aneuploidy was inducing fitness heterogeneity for a variety of stresses, guaranteeing that at least one of the aneuploids would be more adapted to the stress environment than the euploids.
Despite the importance of this pattern to the adaptability, we had no idea to what it was due. Hence, we set off to derive a minimal biologically sound model that could explain the data at hand. Our initial inspiration came from the Weibull distribution (Weibull 1939), which is one of the most common distributions used to describe the non-random failures, commonly used in civil engineering and component quality control. In the search of the fundamental mathematical reasons that would explain why such different processes would follow a common distribution, we came to realize that Weibull was an instance of a general class of extreme values distributions, that were generated by a large class of underlying distribution and mechanisms driving them and did not depend on the underlying distribution, provided some conditions were satisfied. Our second intuition was to look for the classes of distributions that aneuploidy could generate on molecular level, that could impact adaptation to almost all the stresses, regardless of the nature of molecular adaptations required to adapt to them. We noticed that aneuploidy leads to an approximately random perturbation of the proteome and that functional effects of those proteins are due to their participation in complexes and pathways. Given that the Gaussian distributions are additive attractors with respect to summation of values sampled from other distributions, this lead us to suggest aneuploidy was leading to a random, approximately gaussian variation of functional pathway activity levels. The attractor nature of the Gaussian distributions is due to their max-entropy nature for a fixed mean and standard deviation and is commonly introduced in statistics textbooks as a consequence of the Central Limit theorem (CLT). However, there are more relaxed formulations of Central Limit theorem, notably the Lyapunov CTL criterion that could be satisfied by the random pathways perturbations due to aneuploidy in a pathway space. In turn, if we assumed a convex function that mapped pathway deviation from a target combination from an optimum for a given environment to a fitness, as well as assumed that the pathway space could be represented as a scalar space with fitness decaying uniformly according to an L2 (Euclidian) distance, we arrived at a model that could recapitulate the pattern of interest.
The development of this model, as well as mapping of biological assumptions to mathematical formalism as well as the implementation of the simulation and data regression code in collaboration with Dr. Boris Rubinstein were my core contributions to the article that constitutes the core of this chapter.
Robustness and Evolvability of a Heterogeneous Cell Populations
While the model introduced in the Chapter 2 allowed us to explain the experimentally observed relative fitness distributions across stress environments for aneuploid yeast, its relationship to existing models, as well as meaning and influence on the model’s behavior of some of the parameters remained insufficiently explored. In addition, while we suggested that our model could be applied to other biological systems beyond yeast, notably cancer, the reference euploid population was not necessarily clearly defined.
Chapter 3 presents a more in-depth work on exploring the model from a theoretical perspective. Here, we derive a closed formula solutions as well as approximate closed form solutions for the standard deviation and mean of the relative fitness. We as well define the properties of a phenotypical reference population and offer an algorithm describing how to detect a reference population. Here, we explore more in depth how our model behavior depends on the convexity of the function mapping the traits deviation from the environment optimum to the fitness (robustness factor) and establishes the link between our model and the well-known Fisher’s geometric model in population genetics, introduced by sir Ronald Fisher to reconcile Mendelian genetics and Biostatistics in the early 1930th.
The impact of this modelling work however goes beyond the confines of aneuploidy and adaptation to stress. Due to its formal similarity with Fisher’s Geometrical model, our model provides an insight as well for the field of population genetics, especially with regards to measuring the complexity of the phenotypic space experimentally, as well as estimate the robustness of the organism. Due to the heterogeneity of cancer and the importance that this heterogeneity plays in allowing it to acquire drug resistance, our search for a reference population in a cancer cell line population yielded a likely candidate to model a multidrug resistant cancer cell line.
For this article, I have contributed to the article writing, figure generation, conceptualization of the model, review of articles in population genetics, mathematical model formalization, raw data extraction through code as well as invention and implementation of the algorithm for phenotypic reference population detection based on the fitness in a set of environments data. Boris Rubinstein provided help with closed form solution derivation and parts of Mathematica code used for regression. Jin Zhu provided previously unpublished data for non-normalized yeast aneuploid fitness in a large set of environments. The article is in submission and its prior publication is contrary to the journal policies.
Prediction of drug pair forming an “evolutionary trap” for breast cancer
In this chapter, we attempted to apply our mathematical model of adaptation to a concrete rapidly evolving biological system – breast cancer. Our goal is to design an “evolutionary trap”, as described in the chapter 2 – a pair of drugs that would efficiently cover the entire trait space available for the breast cancer cell lines and hence ensure that even multidrug resistant breast cancer cell populations can be targeted and eliminated.
To do this, we rely on the data from the cancer pharmacogenomics screen assay performed and published previously (Daemen et al., 2013; Heiser and Sadanandam, 2012). In their screen, authors use a collection of 70 breast cancer cell lines, which they tested for growth inhibition by a palette of concentration of 90 different therapeutic compounds, ranging from 10e-8 10e-3 molar for 72 hours. Their screen included several lines of non-tumor-inducing immortalized breast epithelia as a reference population, as well as pulled together a highly diverse therapeutic compound set, with vastly different modes of action, for a number of them not even targeting cancer specifically. We were interested in finding the combination of drugs that, at given concentrations, would have had a complementary killing effect on the breast cancer cells, while leaving non-tumorigenic cells grow. To find such a pair of drugs, we made three assumptions. First, we assumed that the breast cancer cell lines, as well as immortalized non-tumorigenic epithelial cell lines responded to the drugs similarly to the breast cancer cells inside patients. Second, we assumed that the breast cancer cell lines were sampling the available trait space in a saturating manner – in other terms they were representing all the space breast cancer cell lines could explore to escape from evolutionary pressure imposed by drugs. Our final assumption was that the action of drugs on the breast cancer for the successive application of two drugs to a cancer cell lines for 72 hours each could be represented as a product of the factor (f) by which the cells grew (f>1) or were killed (f<1) during that period.
First, we needed to process the raw data supplied by (Damien et al. 2013). The data contained a different number of replicates for each compound – cell line pair, as shown in the Figure 2 below. Given that the proxy for cell proliferation and fitness chosen by authors was change in optical density, we needed to account for the acquisition error margin of the instruments. To do this, for each 96 well plate in which the assays were performed, we calculated the Optical Density (OD) of four empty wells. Ideally, their OD would all be set to 0, but in reality, we observed a curve centered about 100 OD units (figure 3, black), that was approximately gaussian (figure 3, green, distortion due to binning). We also noticed that several plates had control wells contaminated, with ODs in the range of several thousands. We discarded those plates from further analysis. In addition to that, we transformed our OD measurements from arbitrary OD units into the units corresponding to the SD of the instrument precision.
Given the number of replicas present for some compound/cell line combinations, our first step was to group them to increase the precision of measurement of the underlying response function. Each plate contained 3 replicas of cell lines for each drug concentration. After calculating the mean and standard deviation of replicas on each plate, we grouped them into single data points, such as visible in figure 4 and then proceeded to grouping replicates from different plates. This presented some challenges. First, some of the replicates presented growth curves showing no difference in the OD between different drug concentrations (semi-transparent data points on the figure 4). We assumed that this was due to those plates receiving a non-working drug sample or incubated in wrong conditions. Due to this, we have eliminated all the drug concentrations that where a difference of less than 10 instrument SD units was observed between minimal and maximal drug concentration. Due to the different starting ODs and batch effect variation between different plates, some of the replicates being as dissimilar in their growth curves as different cell lines, as shown in Figure 5.
Information flow framework for biological network analysis
Our quantitative models explaining the beneficial effect of aneuploidy on the robustness were a promising start. However, we knew that in yeast, adaptation to specific stresses is mediated by aneuploidies with specific patterns. For instance, Radicicol resistance in S. Cerevisiae is conferred specifically by the chromosome 15 gain (Chen et al. 2012). In cancers, specific segmental aneuploidies are associated to specific cancers, such as Chromosome 10 loss to glioblastoma multiforma (von Deimling et al. 1992).
Thanks to the genomic characterization of the breast cancer cell lines throughout the last 15 years, both from the sequencing and DNA CHIPs, we can quantify large-scale segmental aneuploidy in cancer with HMM, as represented in the figure 9 below (code used to generate it available at https://github.com/chiffa/Karyotype_retriever). Once we separate the large-scale segmental aneuploidy in a collection of 53 breast cancer or breast epithelia immortalized cell lines (Neve et al. 2006) from the local amplification, and look at the former (figure 10 below), we can see a specific pattern of common losses and gains of chromosomes.
Table of contents :
Summary:
Acknowledgments
Published content
Chapter 1: General Introduction:
Chapter 2: Targeting the adaptability of heterogeneous aneuploids – Geometrical model of adaptation
Chapter 3: Robustness and Evolvability of a Heterogeneous Cell Populations
Chapter 4: Prediction of drug pair forming an “evolutionary trap” for breast cancer
Chapter 5: Information flow framework for biological network analysis
Chapter 6: Essential genes as evolutionary dead-ends in biomolecular networks
Chapter 7: Experimental investigation of aneuploidy evolvability enhancing potential
Chapter 8: Experimental investigation of aneuploidy impact on intra-nuclear chromosome localization and motility
Chapter 9: ImagePipe: a Python Framework for Biological Microscopy Analysis Pipelines
Chapter 10: Example of application of ImagePipe: import of HS-aggregate related proteins into mitochondria
Chapter 11: General conclusion
Bibliography
