Model genotype-phenotype mappings and the algorithmic structure of evolution

Abstract

Cancers are complex dynamic systems that undergo evolution and selection. Personalized medicine approaches in the clinic increasingly rely on predictions of tumour response to one or more therapies; these predictions are complicated by the inevitable evolution of the tumour. Despite enormous amounts of data on the mutational status of cancers and numerous therapies developed in recent decades to target these mutations, many of these treatments fail after a time due to the development of resistance in the tumour. The emergence of these resistant phenotypes is not easily predicted from genomic data, since the relationship between genotypes and phenotypes, termed the genotype-phenotype (GP) mapping, is neither injective nor functional. We present a review of models of this mapping within a generalized evolutionary framework that takes into account the relation between genotype, phenotype, environment and fitness. Different modelling approaches are described and compared, and many evolutionary results are shown to be conserved across studies despite using different underlying model systems. In addition, several areas for future work that remain understudied are identified, including plasticity and bet-hedging. The GP-mapping provides a pathway for understanding the potential routes of evolution taken by cancers, which will be necessary knowledge for improving personalized therapies.

Keywords: genotype–phenotype mapping; mathematical model; mathematical oncology; personalized medicine.

Figure 1.

Cancer and the biological hierarchy. Genetic alterations induce modified intra-cellular signalling and drive the emergence of cancerous cellular phenotypes. The cells aggregate to form cancerous tissues (tumours) that eventually disseminate through the body. A reductionist approach suggests that this complex system can be understood by considering the basis (genetic) units. This approach fails owing to feedback mechanisms that bridge downward in the hierarchy. Evolution is an example of a such a mechanism as selection at the cellular, tissue or organ level determines which altered genotypes survive. Contrasting the reductionist approach is quantitative holism (qolism). Reproduced with permission from Anderson & Quaranta [38]. (Online version in colour.)

Figure 2.

A graphical representation of Fisher’s geometric model. The genotype space $\mathcal{G}$ and phenotype space $\mathcal{P}$ are both ${\mathbb{R}}^M$ and the GP-mapping is the identity. Mutations are vectors $\mathbf{m}\in {\mathbb{R}}^M$ that are added to a phenotype $\mathbf{p}\in {\mathbb{R}}^M$. The environment is determined by a globally optimal phenotype ${\mathit{\Theta }}_E\in {\mathbb{R}}^M$ and fitness as a function of the distance (z = ‖p − Θ_E‖) of a phenotype from this global optimum. Beneficial mutations are those for which p + m lies closer to Θ_E than p (inside the dashed circle). Deleterious mutations are those that generate a phenotype outside of this circle and neutral mutations those that generate a phenotype an equal distance from Θ_E (dashed circle). (Online version in colour.)

Figure 3.

Fitness landscapes as an evolutionary system. The fitness landscape GP-mapping assumes that the phenotype is a single fitness value. In a schematic view of the landscape, the genotype space is projected into the x–y plane and fitness represented as the height above the plane. This representation emphasizes the dynamics of evolution as an ‘up-hill’ walk. (Online version in colour.)

Figure 4.

Forms of pairwise epistasis. The fitness (height above the plane) effects of different mutations, modelled as flipping a single bit of the genotype, can differ depending on the genetic background. These forms of epistasis can inhibit evolutionary trajectories between two genotypes. (Online version in colour.)

Figure 5.

The RNA folding model. (a) RNA folding as an evolutionary system $\mathfrak{E}$. The GP-mapping takes a genotype g to a minimum free energy secondary structure p through an RNA folding algorithm (ViennaRNA [110]). The environment is specified as a target structure e = p* and fitness is determined as a transformation (w) of the distance (d) between a structure and this target. (b) Phenotypes/secondary structures are topologically defined, thus two RNA strands can fold into the same target structure. (Online version in colour.)

Figure 6.

The stochastic RNA model. Thermal fluctuations cause RNA strands to spontaneously unfold and fold again into different secondary structures. In this model, the phenotype is not fixed but switches between secondary structures with free energy below some threshold. The time spent in each secondary structure is dependent on an environmental factor, the temperature T. This phenotypic switching is an example of bet-hedging in the GP-mapping. (Online version in colour.)

Figure 7.

Schematic of the gene regulatory model for a GP-mapping. The genotype (blue arrows) defines the regulatory actions of genes on one another. The phenotype, p, is determined by the stable state for the network, under the update rule specified in equation (3.10). The initial condition of the network can be taken to be genetic, environmental, or arbitrary, depending on the specifics of each study. (Online version in colour.)

Figure 8.

The fitness function for the GRN model. A metric value (in this case, Hamming distance) between a phenotype, p (red), and a target phenotype e = p* (environment, purple) is calculated and a transformation of this value is used to determine the fitness, f(p, e), of an individual with phenotype p in the environment e. (Online version in colour.)

Figure 9.

The neural network model and phenotypic plasticity. (a) Schematic of the (feed-forward) NN model. Environments, e = (e₁, e₂, e₃), form the input (purple), the genotype determines the internal weights of the NN (blue) and the phenotype (red) is computed by evaluation of the NN. (b) The traditional (linear) representation of a reaction norm mapping an environment, e, to a phenotype p. Genetic variation results in changes to this relationship. (c) A higher dimensional and nonlinear reaction norm, these more complicated forms of the reaction norm can be represented by the NN model. (Online version in colour.)

Figure 10.

The CRN model for a GP-mapping. (a) A schematic for the CRN model. Genotypes encode initial values for expressed molecules, x and y. The GP-mapping is encoded as a bistable chemical reaction network that is stochastically simulated to determine the phenotype, A or B. (b) Mutations in the CRN model are alterations to the initial abundance of the intra-cellular molecules. (c) The environment is represented as a binary variable which could for example indicate the presence or absence of a drug. A fitness value is chosen for each combination of phenotype and environment. (Online version in colour.)