Epigenetic Entropy: Yardstick for Measuring Cancer Risk

An earlier post from March described the epigenome, the set of modifications of DNA and chromatin which provide control of gene regulatory networks needed to guide cellular differentiation during development and then maintain stable cellular phenotype in mature tissues. The epigenome, visualized in cell microscopy as nuclear euchromatin and heterochromatin, are chromatin states which either permit or inhibit ‘reading’ of genetic information through control of the access of polymerase enzymes to conduct DNA transcription. This hierarchical control over gene expression affecting the accessibility of RNA polymerases for transcription of specific sets of genes termed a gene network and acts to establish cell phenotype. This regulation allows for flexibility in phenotypic expression permitting the reversal of cell fate from a differentiated towards an undifferentiated state needed for example in tissue response to injury, termed lineage plasticity. Plasticity is a key concept in understanding normal development and tissue repair. Unfortunately, it also provides an opening for cancer development.

When we think about cancer evolution, we usually consider a process of clonal evolution, the emergence over time of increasingly diverse subpopulations of malignant cells driven by gene mutations and chromosomal rearrangements, sometimes in response to environmental selection pressures, leading to a cardinal feature of cancer, tumor cell heterogeneity. There is however an alternate means for generating tumor heterogeneity which may result through the development of increasing cellular plasticity, the ability of tissues to adopt new, and in some circumstances malignancy-defining phenotypes leading to the same result of cellular variability. Andrew Fineberg of Johns Hopkins University and Andre Levchenko of Yale have related this loosening in normal control of cellular plasticity as a product of dissipation in the predictability of epigenetic regulation accounting for the disruption of that regulation. That is, increasing phenotypic plasticity is a result of increasing epigenetic entropy.

The consistency of the epigenome in mediating regulation of gene network activation is subject to random perturbations, stochasticity, the inevitable loss of information which occurs during signal transfer not unlike the static you might encounter during a telephone call. The epigenome is also subject to cell intrinsic influences such as mutation of epigenetic modifiers, the genes whose products directly methylate DNA or ligate chromatin. Importantly, the epigenome is also subject to extrinsic influences from such factors as aging, chronic inflammation and environmental exposures which modulate that control indirectly by affecting the upstream focus of the epigenetic modifiers or alternatively the downstream gene products which mediate epigenetic control. The emergence of this epigenetic variability can be measured using a concept from thermodynamics, entropy, adopted by information theory as a gauge of the predictability of accurate information transmission. As consistent epigenetic governance over gene network activation becomes undermined, the likelihood grows for the inadvertent opening of otherwise hidden gene programs leading to the increasing phenotypic plasticity characteristic of cancers.

If we accept that higher levels of entropy of epigenetic regulation account for cancer phenotypic plasticity, then a measure of this parameter would likely be useful in understanding propensity for cancer emergence and evolution. To do this Fineberg and Levchenko have employed another concept from physics, quasi potential energy (quasi meaning resembling the physics concept of potential energy) to create either an epigenetic or a gene expression landscape (Science 379, 552, 2023). The idea for a landscape model to explain development was first argued by embryologist Conrad Waddington to conceptualize the multiple pathways embryonic cells might traverse to establish differentiated cell subtypes. Presently, to describe these developmental pathways, contemporary developmental biologists have had to rely on Ordinary Differential Equations to model these landscapes comparing the rate of change of phenotype as a function of the concentration of one or another relevant biomolecule. Such calculations, however, broke down when molecular concentrations were low if they were measurable at all. Adjustment factors in these calculations became necessary to account for the level stochasticity occurring in such a relationship but then having the effect of making the equations become less solvable.

More recently a means of circumventing this obstacle for quantitating phenotypic expression comes from progress in molecular biology which now permits geneticists to sequence the RNA molecules of individual cells. Achieving single cell molecular level quantitation can then be represented as a map of the chances and degree of RNA expression across an entire range of cells comprising a tissue or a cancer. Single cell sequencing to characterize the chances a particular phenotype being present in relationship to a parameter of interest allows the construction of a probability distribution. Taking advantage of another relationship from thermodynamics, that the probability of finding a molecule in one particular state among others varies inversely with the energy level accompanying that molecule, the Boltzmann Distribution. Ludwig Boltzmann was an Austrian physicist of the 19^th century who employed the language of statistics to describe the mathematical relationship between the energy of a system and that system’s entropy, what has been termed statistical thermodynamics. Said most simply, the macroscopic features of a system, for example in this case, cell phenotype, are the sum of the random fluctuations of the microscopic constituent of that system, here the gene regulatory network.

Figure 1

Using this relationship between the probability of a state being present and its accompanying potential energy level, Fineberg and Levchenko describe a means of ‘translating’ a map of a probability distribution of a certain phenotype into a quasi-potential energy landscape. In such a landscape, as diagrammed in figure 1A, the phenotype with the highest chances for occupancy will be represented by the lowest quasi-potential energy. These quasi-potential energy ‘wells’ can then be thought of as attractors, phenotypic states with the highest likelihood of occurrence, exerting a ‘pull’ on the surrounding landscape. This would be similar to a ball rolling along the surface of an analogous physical landscape which would be ‘attracted’ to the bottom of the landscape well, corresponding with the lowest level of gravitational potential energy.     

An advantage in using a landscape representation of phenotype as a quasi-potential energy landscape in relationship to either an epigenetic configuration or associated protein products of a gene expression network is that it permits a means of quantitating the entropy of that epigenetic state or gene expression profile. From Boltzmann, the entropy of that representation can be measured based on the depth and width of the accompanying attractor. This approach allows investigators to sidestep the earlier problem of how to account for the stochastic ‘noise’ of a system which had complicated earlier calculations. In this model, tissues characterized by a deep and narrow attractor, representing a low level of epigenetic or network entropy would allow relatively little fluctuation of defining biomolecules and thus would make unlikely an ‘escape’ through stochastic molecular fluctuation into an alternate, adjacent phenotypic states. The plasticity of that tissue would be low, typical of healthy, mature tissue as seen in Figure 1B. If on the other hand forces were to occur, affecting epigenetic regulation of that tissue, interfering with the regulatory supervision of gene expression, then a more epigenetically variable environment might result causing a wider distribution of molecule concentrations associated with that phenotype reflecting the higher level of network or epigenetic entropy. The landscape attractor would now seem molecularly less attractive.

Figure 2

Another advantage of the landscape model as seen in the diagram in Figure 2 permits a more ready visualization of the effect of this higher entropy level on the probability of phenotypic transition. As an attractor loses quasi-potential energy, the depth of the landscape attractor lessens, and the cells of that phenotype become less constrained by the attractor. Through stochastic fluctuations of phenotype defining molecules, the chances for cells within the attractor to transition to other adjacent attractors representing different and possibly malignant phenotypic state becomes more likely. As a metaphor for understanding this, if you were living at the bottom of a steep valley, it would take considerable expenditure of energy to allow you to leave. If the valley walls, however, were shallower, or if you discovered a ‘pass’, then the journey to explore other adjacent valleys would seem less prohibitive, that is, it would take less energy to accomplish. And once you had crossed over into the new valley, you might find yourself well suited there and decide to stay. In cellular terms, an epigenetically variable landscape with high entropy represents a permissive state increasing the odds that a deleterious phenotype might emerge as illustrated in Figure 1C.

Figure 3

There is a close and interdependent relationship that can be detected between the three classes of epigenetic regulatory influences, the epigenetic modifiers which directly act on DNA and chromatin such as DNA methyltransferases or histone demethylases, the modulators such as signal transduction pathway kinases such as MAPK and NOTCH and the mediators of the epigenome such as transcription factors are illustrated in Figure 3. The profile of a gene regulatory landscape mirrors that of the associated epigenetic landscape, both of which are subject to external modulating influences. A prime example of this relationship is the transcription factor p53 which ordinarily acts as a tumor suppressor regulating cell division but alternatively may also act to affect levels of activity of certain DNA methylases and demethylases. Loss of p53 function as would occur through the effect of mutation would release that inhibition resulting in increasing epigenetic entropy. Similarly, epigenetic modulators, sensitive to both developmental influence as well as external influences from the environment or from inflammation can affect both profiles to increase entropy. Other examples of this connectivity include aging or gene and protein interaction networks both of which may serve to increase epigenetic and gene regulatory entropy. Stepping back to encompass the big picture of epigenetic regulation, the DNA/chromatin modifiers, the signal transduction pathways, the gene transcription factors and the external environment are all interacting across this broad framework to adjust cellular phenotype.

Figure 4

The authors describe the possibility of using these landscape representations in conjunction with other maps of phenotypic expression for the possibility of designing specific therapy. An example of this strategy illustrated in Figure 4 is a mapping of the phenotypic landscape of cells in response to the loss of imprinting (LOI) of the Insulin-like Growth Factor-2 (IGF2) gene, onto the accompanying plane of corresponding Bcl-family proteins. Imprinting is a key developmental process in which one of two parental alleles of a gene are silenced through epigenetic mechanisms. LOI may occur either based on an inherited syndrome or as an acquired trait. Bcl proteins act to regulate apoptosis, a controlled means of cell death by which tissues manage organ homeostasis. BAX proteins promote apoptosis while Bcl-2 proteins inhibit it. Loss of imprinting of the IGF2 gene leads to a constitutively high level IGF2 causing an imbalance in the ratio of downstream activated signal transduction pathways of the Erp and Akt kinases, illustrated in Figure 4A. That shift, represented by a transition in the landscape of phosphorylated (and thus active) Erkpp and Aktpp which can then be mapped onto the BAX/Bcl-2 phenotype plane demonstrated in Figure 4B. In that mapping, the LOI phenotype maps more closely to the boundary between cell death or survival determined by the ratio of BAX and Bcl-2 levels when compared with that of wild type phenotype. A therapy approach then such as an inhibitor of the IGF receptor, by shifting this mapping to the left, an indication of lower kinase activation levels, would have a selective effect on chances for survival of LOI affected cells compared to the fate of wild type cells. Using this technique of applied epigenetics then would allow clinical investigation taking advantage of the knowledge of a vulnerability of the LOI cells which otherwise might not have been recognized.

Summarizing, from breakthroughs in molecular biology, new quantitative methods from the application of statistics and (information) entropy provide a basis for understanding a remarkable aspect of developmental biology, phenotypic plasticity, the ability of a cell to reverse differentiation to regain pluripotency for purposes of tissue regeneration and repair from injury but at the same time leaving that tissue vulnerable to dysregulated epigenetic and gene regulatory control which in turn increase chances for the emergence of maladaptive and deleterious cell subtypes. In an upcoming third post on epigenetics in May, this landscape model of phenotypic expression will be used to further explore the emergence of other characteristic features of cancer, the cancer hallmarks in the context of an additional force shaping cancer evolution, that of selection.

James Cunningham 

Legends for Figures 1-4 Taken from Feinberg et al., Science 379, eaaw3835 (2023) 

Fig. 1. Gene expression and epigenetic landscapes control normal and cancer cell functions.

(A) Gene regulatory networks and availability of genes for expression can define the probabilistic distributions of proteins expressed within a cell population. In this example, the network of interacting proteins that includes the molecules A and B (top) and the underlying epigenetic control determining the availability of the corresponding genes for expression define the distribution of the expression of molecules A and B (middle). This probability distribution can be experimentally measured and converted into a gene expression landscape by calculating the corresponding quasipotential distribution (bottom) (see Box 1). The epigenetic landscape can be similarly determined by experimentally measuring the probabilistic distributions of epialleles, measuring DNA methylation marks at specific loci, or by performing other measurements of epigenetic regulation across populations of cells and tissues and then also converting these probability distributions into corresponding underlying quasipotential landscapes. The landscape analysis allows conceptual accounting for abundance and dynamics of molecular species, shown here as a trajectory of a particle inside a quasipotential well, with the particle position defined by the current concentrations of A and B that can change probabilistically in time, with the quasipotential wells interpreted as the landscape attractors. (B) Various scenarios of landscape alterations and the corresponding changes in the molecular distributions, shown as joint distributions of the molecules A and B and the corresponding entropies H1 to H5. Implementation of these scenarios in the context of carcinogens is extensively illustrated and discussed in the text. Oncogenic mutations of epigenetic modifiers and modulators or environmental inputs can lead to the formation of new stable attractors with the overall entropy H2 greater than the original entropy H1 (H2 > H1), generating phenotypic heterogeneity (input 1′) or, alternatively, enlarging the existing attractor with the new entropy H3 > H1, generating a more plastic state (phenotypic plasticity), with cells capable of stochastically and dynamically exploring this attractor and thus transiently adopting different phenotypes. In both cases, entropy increases versus H1 and it is possible that H2 = H3, thus making entropy less discriminating than the full landscape picture in the analysis of cell states. These new landscapes can be further altered by oncogenic and environmental inputs, so that one of the attractors becomes dominant (input 2′), associated with a lower entropy value (H4 < H2) or, alternatively, with the narrowing of the wider (and more plastic) attractor (input 2; H5 < H3). Again, it is possible that H4 = H5, requiring the landscape analysis rather than entropy analysis alone for full characterization. The narrowing of the wide attractor because of either environmental or intrinsic inputs (input 2) is frequently reversible and context dependent, further elaborating the more plastic overall state (transient nature of input 2 described by a bidirectional arrow). The transiently occupied attractors can be simultaneously occupied by distinct cells in the population. Small arrows correspond to stochastic fluctuations of molecular concentrations within individual attractors. (C) Gene regulation and epigenetic landscapes of cancer cells can be complex and have multiple attractors, corresponding to distinct and stable cell states and phenotypes, which may be reshaped by oncogenic mutations, cell aging, environmental inputs, and other perturbations, leading to mutual accessibility of the attractors, more plastic cell states, and an increase in the phenotypic plasticity.

Figure 2:

Epigenetic landscapes and phenotypic plasticity in cancer.

Regulatory networks can define the number and probabilities of stable cellular states adopted by a cell population, representing attractors in the epigenetic landscape. Diverse inputs can promote transitions (and corresponding phenotypic plasticity) between cellular states within landscapes corresponding to the normal tissue (fewer attractors) and cancerous tumors (emergence of new attractors), as defined by parameters P1 and P2 that correspond to effective concentrations of landscape-defining molecules.

Fig. 3. Interplay between epigenetic and gene expression landscapes.

Developmental and environmental factors and genetic mutations can affect diverse modulators of epigenetic control and gene expression, such as signaling and cell communication networks, frequently leading to diversification of cell states. These modulators may directly affect mediators of epigenetic states, such as DNA demethylases, and gene expression, such as transcription factors, which also can directly interact with each other. Examples of these molecular regulators discussed in the text are shown here. The result is alterations of the epigenetic and gene regulation landscapes that are tightly coupled, for example through the action of mediators of epigenetic control, influencing accessibility of genes for regulation, and the magnitude and variability of gene expression. Certain additional inputs may be more specific to each of the landscapes, such as the epigenetic drift with cell aging primarily leading to a widening of the landscape attractors, higher plasticity and higher entropy of the state, or protein-protein interaction and gene regulatory networks, stabilizing various attractors and serving to decrease the plasticity and entropy.

 

Fig. 4. Connection between an epigenetic landscape and variable phenotypic outcomes.

(A) An epigenetic alteration—LOI of the IGF2 gene, implicated in Wilms tumor, doubling the signaling input—can lead to rewiring of the signaling network activated by the IGF2 receptor IGF1R (depicted as IGF1Rp) through altered receptor trafficking (IGF1Rint), degradation (ϕ), and altered balance of activation of the downstream signaling pathways activating Erk (Erkpp) and Akt (Aktpp) kinases. Rebalancing of Erk and Akt activities translates into transcriptional up-regulation of IGF1R and a higher proliferation rate but also rebalancing of pro- and antiapoptotic protein abundances (BAX versus Bcl-2, respectively), leading to an increased propensity for cell death (108). The integral signs represent integration over time of signaling activities. (B) The landscape alterations that correspond to a change in phenotype (top) are the altered expression and activity of signaling pathway molecules (and thus gene regulatory landscapes in the bottom panel) in response to alteration of epigenetic landscapes (IGF2 LOI). This leads to emergence of a new attractor in addition to the WT attractor, resulting in a mosaic WT-LOI cell distribution in the tissue. This landscape alteration can be mapped onto, for example, the apoptosis phenotype-defining network by a quantitative analysis of the dependence of the BCL family protein distributions on the signaling inputs, thus enabling a direct translation of the landscape alterations into phenotype distributions. In this example, the mapping can be visualized as WT and LOI cell distributions mapped with respect to the areas of cell survival and death on the (BAX, Bcl-2) phenotypic plane, which suggests how treatments targeting LOI cells may be developed to spare the WT cells. Arrows in the lower panel represent the effect of drugs, such as IGF1R inhibitors, shifting the landscape and phenotypic distribution toward the boundary separating survival and death, with the red areas depicting the effect on the WT and LOI cell populations. (C) A more general view of landscape mapping onto the apoptosis phenotypic plane. By analogy with Fig. 1B, one can contrast mapping of a large attractor versus two more limited attractors, representing the difference between a plastic and stochastic state (phenotypic plasticity) versus a state with two alternative stable attractors (phenotypic diversity). The more plastic state can allow cells to escape from the death area to the survival area even in the presence of a treatment [such as in (B)], by stochastically exploring the available attractor, whereas a combination of more stable attractors (with the same overall entropy as the more plastic state) can allow for selective targeting of one but not the other attractor. Therefore, the treatment strategy suggested in (B) may benefit from the initial intervention, stabilizing smaller attractors within a larger one and thus decreasing the plasticity of the state, particularly through epigenetic perturbations.