Non-equilibrium statistical physics, transitory epigenetic landscapes, and cell fate decision dynamics
NNon-equilibrium statisticalphysics, transitoryepigenetic landscapes, andcell fate decision dynamics
Anissa Guillemin & Michael P.H. Stumpf , School of BioScience, University of Melbourne, Melbourne, Parkville 3010, VIC, Australia School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, [email protected]
Abstract
Statistical physics provides a useful perspective for the analysis of many complex systems;it allows us to relate microscopic fluctuations to macroscopic observations. Developmentalbiology, but also cell biology more generally, are examples where apparently robust behaviouremerges from highly complex and stochastic sub-cellular processes. Here we attempt tomake connections between different theoretical perspectives to gain qualitative insightsinto the types of cell-fate decision making processes that are at the heart of stem celland developmental biology. We discuss both dynamical systems as well as statisticalmechanics perspectives on the classical Waddington or epigenetic landscape. We find thatnon-equilibrium approaches are required to overcome some of the shortcomings of classicalequilibrium statistical thermodynamics or statistical mechanics in order to shed light onbiological processes, which, almost by definition, are typically far from equilibrium.
Cells are often seen as the fundamental level from which to start investigating biological systems [1]:drill down in detail, or increase resolution, and we end up with intricate molecular processesand arrangements that shape the cells physiology and behaviour. Zoom out, and we observehow cells interact with each other, compete for resources, or self-organise into tissues and wholemulticellular organisms [2, 3]. Genes and their gene products—mRNA and proteins—can onlyfulfil their evolved biological function in the context of cells and organisms [4]; and for complexmulticellular organisms we cannot ignore the discrete cellular structure of tissues.Cells are themselves highly dynamic and dynamically changing systems. They sense and respondto changes in their environment; they control their internal state, adapting it where necessary;and many cells coordinate their reproduction into pairs of daughter cells. In multi-cellular,but also in some uni-cellular organisms, this process can involve profound changes in the cells’characteristics and states.Dynamical systems describe how many natural, including biological, systems change over time.Analysis of ordinary differential equations (ODE) describes the behaviour over time of e.g.,predator prey systems, biochemical reaction systems, neuronal activity, or physiological processes.If x denotes the state of the cell (abundances of mRNAs, proteins, lipids etc) then we assume Page 1 of 14 a r X i v : . [ q - b i o . CB ] N ov that we have a (high-dimensional) function, f ( x ), such that the rate of change in x is given by dxdt = f ( x ) (1)with “cell states”, x ∗ , corresponding to the locally stable stationary points of the dynamicalsystem, calculated from f ( x ∗ ) = 0 , that is those solutions for which the eigenvalues of the corresponding Jacobian (assuming linearstability analysis suffices), J = (cid:18) ∂f i ( x ∗ ) ∂x j (cid:19) i,j =1 ,...,d are all less than zero [5, 6]. Where linear stability analysis is not sufficient to assess the (local)stability, other, more involved methods need to be invoked.The stochastic differential equation (SDE) counterpart to Eq (1) takes the form dx = f ( X ) dt + g ( X ) dW t , (2)where g ( x ) captures the functional form of the stochastic contribution to the dynamics, and W t is a Wiener Process increment. The stability of stationary points of deterministic dynamicalsystems under the influence of stochastic dynamics is in general difficult to assess analytically [7,8].Instead, in most cases, approximations or simulations are required.What we are interested in are new ways of determining and analysing the functional form of f ( x ) (and g ( x )). This would open up the possibility of making more and better mechanisticmodels in cell biology. With the exception of a handful of simple models of e.g., embryonic[9–12] and haematopoietic stem cells [13], we have precious few mathematical models of therelevant differentiation systems. Analysis of data is thus largely descriptive, but even fromthese descriptions we can learn or distill some important lessons that could inform mechanisticmodelling in the future [14]. Three such examples include molecular noise, the dynamics ofgene regulation, and the time it takes for cell-fate decisions to take place. First, noise in geneexpression, or cell to cell heterogeneity, appears to be closely associated with the transitionbetween cell states [15–17]. Second, there is clear evidence that the regulation at the geneexpression level is highly dynamic and shaped by factors at the epigenetic, transcriptomic,proteomic, and post-translational modification levels [18–21]. We cannot describe this in termsof static gene regulatory networks, and instead need to develop explicitly dynamical descriptions;even then we need to take into account the uncertainty in these networks [22]. Third, the timingof a transition appears to indicate that differentiation is a non-Markovian process [23].In the following we will outline a set of qualitative frameworks for the analysis of cell differentiationdynamics, developing their connections, and follow one of these, non-equilibrium statisticalmechanics , further in order to characterise transitions between states or cell fates. The theory of dynamical systems offers a set of tools that allow us to investigate developmentalprocesses. There is already a set of well studied problems, including different stem cell differenti-ation systems [9–13], segmentation in insect development [24], neural tube formation [25], andTuring patterns [26, 27]. But for the vast majority of systems of concrete biological interest welack such mechanistic descriptions.In addition to developing these models from the bottom up, we can also take a more abstractperspective, again grounded in the theory of dynamical systems. Below we outline two suchapproaches.
Cusp Catastrophe
Fate 1Fate 2Stem Cell D i ff e r e n t i a t i o n C e l l - t y p e s p e c i fi c Figure 1:
Example of a Cusp Catastrophe and its relationship to developmental dynamics. The twocontrol variables are a differentiation marker, which leads to loss of stem-like properties,and factors affecting cell-identity, such as the relative abundance of competing transcriptionfactors.
If we identify cell fates with the stationary points, x ∗ , of a dynamical system, then, even if we donot know the structure and form of the dynamical system, i.e., we do not know the mathematicalform of f ( x ) in Eq (1) or Eq (2), we can still make some general qualitative statements. Work inthis area, especially by Ren´e Thom [28], was in fact partly inspired by problems in developmentalbiology. Catastrophe Theory [28–30] was developed to characterise the qualitative behaviour exhibited bydynamical systems as system parameters change. A central tenet of this approach, borne outempirically as well, is that in many cases even high-dimensional systems can be understood interms of dynamics of a much lower dimensional system. Here the evocative term “catastrophe”refers to a sudden change in the qualitative nature of the set of solutions of a dynamical system,caused by a smooth or small change in some model parameter, system variable, or control input.In the example in Figure 1 such a change is observed in the number of fates that are accessible,as cell-type specific markers are varied. There are regions in parameter space where either one ofthe fates (Fate 1 or Fate 2) is realised, and regions where both can co-exist. As the differentiationfactor is reduced, only a single state, the stem-like state, can exist. In the language of catastrophetheory this is an example of a cusp catastrophe.Despite the appeal of this framework there are considerable limitations, including the fact thatmany of the theoretical results are limited to gradient systems, although stability propertiescan apply more widely [31]. Certainly a central cornerstone of catastrophe theory, structuralstability [29], will have wider implications. We mean by this that the qualitative behaviour ofa mathematical model or theoretical system remains stable even when the model is changedslightly. Such structural stability of a model would be desirable, as we know that real-worldsystems differ from our models, and often quite considerably so. Clearly we should thereforestrive in our modelling to only consider structurally stable systems.
Following Waddington’s groundbreaking conceptual work—which has had great influence onRene Thom’s work [28]—the well-known epigenetic landscape was long primarily seen as a usefulmetaphor for developmental processes [32]. In the 21st century, however, it has increasinglybeen seen as a computational tool in its own right [33–39]. And there is a rich mathematicalliterature to draw on, that has only rarely been tapped into so far [30], notably related to Morsetheory [5, 40].Here, however, we follow in the first instance the statistical physics perspective developed above.We shall also restrict ourselves to gradient-like systems, i.e., where f ( x ) in Eqs (1) and (2) canbe written as f ( x ) = −∇ U ( x ) , (3)where U ( x ) is a potential [5] (or quasi-potential [41] under suitable circumstances). For stochasticsystems (for deterministic systems, (1) resulting densities are typically sums of Dirac δ functions),we have for the probability density of the solution of Eq (2) to be at xdx , π ( x ) = exp( − U ( x )) , (4)details to this can be found in [35–37].Local minima of the potential correspond to (locally) stable attractors of the dynamics and canthus be related to cell states [33,42]. And Morse theory [29,43], which applies to gradient systems,allows us to relate the different fixed-points of gradient systems in the case of deterministicdynamics. Notably, any set of locally stable fixed points, is separated by (at least) one saddlenode, i.e., a fixed point where the Jacobian of the potential, U ( x ), has both positive and negativeeigenvalues [5, 29]. A different approach is rooted in theoretical physics and has found widespread use in e.g.,ecology [44], signal transduction [45–47], and gene regulation [48, 49], but it has also been shownto be helpful in developmental and stem-cell biology [12, 23, 50, 51]. Statistical physics linksmicroscopic behaviour of e.g., molecules or atoms, to macroscopic observables such as pressure.In the context of cell biology, the precise molecular composition characterise microstates , whereasthe cell-types are the observable macrostates . Just as in statistical physics, each macrostate isassociated with a large number of corresponding microstates. In biological terms, the microstatesassociated with a given macrostate represent all molecular configurations (chromatin states,mRNA and protein concentrations, transcription factor activities etc) corresponding to a givencell state (Figure 2).Statistical mechanics is concerned with the long-term behaviour of a system [52, 53], in particularassigning probabilities of different macrostates being realised. We will start by defining someof the terminology. First, we use Latin and Greek letters to denote micro- and macrostates,respectively; here and below we follow the exceptionally clear terminology set out by Attard [53].We assume that we can meaningfully define weights for the different microstates, ω i . We thenhave for the weights of the corresponding macrostates, ω α , ω α = (cid:88) i ∈ α ω i . (5)Summing over all microstates and macrostates must necessarily give the same value, W , that iswe have (cid:88) α ω α = (cid:88) i ω i = W, (6)where W is the total weight (to be made precise below). For notational convenience we use thesum symbol, (cid:80) , rather than the integral (as would be more appropriate for continuous statespaces). From the weights we obtain the probabilities of micro- and macrostates, p i = ω i W , and p α = ω α W , (7) G G G G G G G G ~G G ~G G ~G G G Figure 2:
Representation of the definition of macrostates and microstates for the case of a small genenetwork representing the phenotype. Macrostate A is represented here by the top network;macrostate B by the network at the bottom. The macrostates correspond to distributions overdifferent microstates. The microstates here are interpreted as the (joint) expression levelsof genes, G (gene 1), G (gene 2) and G (gene 3). The expression levels of G overlapbetween the two macrostates, but the ( G , G , G ) microstates associated with the macrostatesare distinct. with which we can, for example, obtain expectation values for functions, e.g., across a macrostate,by averaging E [ r ] α = (cid:88) i ∈ α p i r i . Here r could denote cell size, or gene expression level associated with a given cell type, if α refersto cell types. Higher moments, including variances are calculated similarly.The entropy plays a central role in statistical mechanics, and here is defined as S = log( W ) , (8)or, if we consider the entropy of a macrostate, as S α = log( ω α ) = log (cid:32)(cid:88) i ∈ α ω i (cid:33) . (9)If we have a uniform distribution over the microstates we can assign unit weight to each microstate, ω i = 1 , ∀ i . The entropy of a macrostate is then simply, S α = log n α , where n α is the number ofmicrostates corresponding to macrostate, α .One of the central results of statistical mechanics is encapsulated in the second law of thermo-dynamics, which states that entropy never spontaneously decreases [52, 53]. Thus spontaneouschange will only ever occur if a change in state leads to an increase in entropy. So in this picture,a change from state α to α (cid:48) will only be observed if S α (cid:48) > S α ; or, in the case of uniform weightover microstates, if n α (cid:48) > n α . We next link the epigenetic landscape with entropy, first by considering equilibrium statisticalmechanics, then by considering transitions between states. We start by developing the totalentropy of the system, Eq (8), further, S = log( W ) = (cid:88) α p α log (cid:18) ω α Wω α (cid:19) (10)= (cid:88) α p α S α − (cid:88) α p α log p α . (11)Here the second term is the conventional representation, which captures the uncertainty associatedwith respect to which macrostate, α , the system is in. The first term captures the internaluncertainty associated with the macrostates; this is often ignored because in many applicationsonly differences in entropy matter, and the whole term can then be viewed as an additiveconstant. But, for example, when considering a system coupled to a reservoir, this term doesmatter profoundly. It is also important for the case where we consider different macrostates,which below include alternative definitions of cell types. Identifying macrostates for cell biology is, perhaps surprisingly, non-trivial. Microstates areversions of gene expression states, associated to a macrostate [23,50,51]. Many of these microstateswill never be attained [36]. We assume that the whole state space can, in principle, be specifiedand we denote it by Γ. The whole set of macrostates, called a collective , has to cover all potentialstates in a non-overlapping manner, that is (cid:91) α α = Γ = (cid:91) i x i and α ∩ α (cid:48) = ∅ . (12)The second condition is typically easy to meet, the first is slightly more problematic: we haveto assign microstates that are potentially never observed [36] to appropriate macrostates. Wediscuss an almost certainly incomplete list of suitable macrostates for cell biology in the following. Phenotypic Definition:
If we have a set of objective phenotypic markers (morphology or cellsurface marker), S = { s , s , . . . , s C } , we may use this as a base from which to define macrostates; α . We then, have, however, three types of microstates: (i) microstates that are observed inthese cell-types; (ii) microstates that are never observed; these have weight ω i −→ a priori complicated: they may correspond to new cell-types or sub-types; theymay correspond to intermediate cell states [14, 54]; or they may be fleetingly visited as cellsexplore the molecular states available [36]. These states need to be considered with some care ifwe want to base macrostates on phenotypic cell definitions. Data-Driven Definition:
Alternatively, observed microstates can be subjected to statisticalanalysis, perhaps, unsupervised learning to group them together and then assign macrostates toclusters [14, 15, 55]. The ambiguity of clustering [56]—especially whether to lump small clusterstogether, or split larger, more extended clusters—is, of course, encountered in this approach.But because of the practical irrelevance of microstates that are never encountered (see above)this approach seems sensible, and unproblematic.
Dynamical Systems Definition:
We can use ideas from the theory of dynamical systems [5].For the deterministic case we can group all microstates, x i which, for t −→ ∞ go to the samestationary state into the same macrostate. This definition assigns every point in Γ to one andonly one macrostate. However, generalisation to stochastic dynamics is not straightforward;furthermore, it does not capture the role of saddle node fixed points, which may play an importantrole in defining intermediate cell states [14]. Mixed Macrostates:
The advantage of the form for the entropy given in Eq (11) is that wecan combine different collectives of macrostates, here denoted by α , β , and γ . We can calculatethe weight of such mixtures using the usual laws for joint probabilities, e.g., we have ω ( αβγ ) = (cid:88) i ∈ α ∩ β ∩ γ ω i . With this, it becomes possible to combine the macrostate definitions above and overcome theirindividual limitations. We can also, through simple relabelling of macrostates, simplify thenotation and have a single subscript to denote the new “mixed macrostate”.
One problem related to the difficulty in developing a statistical mechanics for stem cell biology,comes from the fact that much of the appeal of statistical mechanics lies in the fact thatentropic arguments can be used to determine (most probable) system states. The second lawof thermodynamics, in particular, states that entropy never decreases spontaneously, and thatthe maximum entropy state is the one realised with high probability [52, 53]. If the macrostatesare not coherently defined then entropy and (cid:80) i ∈ α p i log p i cannot be used to assign the mostprobable states.Reports, for example, that entropy across cell-populations becomes maximal around the transitionstate are thus not necessarily in violation of thermodynamics [57]: clearly the most probablestates (i.e., the states with highest probability as t −→ ∞ ) will correspond to fixed points andtheir vicinity. High entropy at transition states could either reflect poor definitions of cell states;or this could simply reflect that the concepts from equilibrium thermodynamics and statisticalmechanics are of limited use in this regime.Both explanations seem eminently plausible, and non-equilibrium statistical mechanics andthermodynamics may offer solutions to the second problems, in particular. We will sketch outtwo such solutions: a brief introduction to transition probabilities between states and how we canuse them to reconcile some of the experimental results. Finally, we briefly turn to consideringdynamic epigenetic landscapes. In non-equilibrium theories we consider explicit change over time. One convenient way is toconsider transitions over time τ , j τ −→ i , as states of interest. We have for the weight of amicrostate j ω j = (cid:88) i ω ( j, i | τ ) , (13)that is the weight of a microstate, j is equal to the sum over the weights of transitions out of j into any other microstate. The weight of a transition between (suitably defined) macrostates β −→ α (where α and β can be in the same or different collectives) is then given by ω ( β, α | τ ) = (cid:88) i ∈ α (cid:88) j ∈ β ω ( j, i | τ ) . (14)Crucially, even if transitions between microstates were deterministic, transitions betweenmacrostates will still be stochastic, because specification of macrostates, α and β , does notprecisely define the start and end microstates, i ∈ α and j ∈ β [53].In order to normalise the weights, we have to sum over all states, or equivalently, all transitions Di ff erentiation factors increasing Figure 3:
The epigenetic landscape, and our quasi-potential representation thereof, will change over timeand in response to the cellular environment. This is illustrated here, where the progressionalong a developmental trajectory leads to the dissolution of old and the creation of newpotential minima aka cell types. Whether this evolution of surfaces is necessarily smooth isnot clear. that can occur, and we get W = (cid:88) i ω i = (cid:88) i,j ω ( j, i | τ )= (cid:88) β ω β = (cid:88) α,β ω ( β, α | τ ) . (15)With this we get for the probability of a transition from state i/α to state j/β to occur at sometime in the future τ , we have p ( j, i | τ ) = ω ( j, i | τ ) W and p ( β, α | τ ) = ω ( β, α | τ ) W . (16)We then have for the conditional probability of ending up in state β given that the system startsin state α p ( β | α, τ ) = ω ( β, α | τ ) ω ( α ) . Now analogously to Eq (9) we can define a new entropy for the transitions between macrostates, S (2) ( β, α | τ ) = log ω ( β, α | τ ) , (17)which we refer to as the second entropy. The advantage of this formalism is that for non-equilibrium systems and equilibrium systems alike, the most likely state transition, given thecurrent state, β can be obtained by determining the state ˆ α ( τ | β ) which maximises the secondentropy, ∂S (2) ( β, α | τ ) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α =ˆ α = 0 . This shifts the focus of the analysis from states to transitions, and may offer a better perspectiveon cell differentiation than a conventional equilibrium statistical mechanics perspective.
This brings us to the final point, and one that may have been implicitly presaged by Waddington[32], and is also apparent in the work of Thom [28] and others (see [30] for a recent review):the epigenetic landscape should not be viewed as a fixed object in time, but one which changesdynamically [36]. For mathematical models of developmental systems—typically of the essentialgene regulation networks—it is possible to calculate the corresponding epigenetic landscapesas quasi-potentials, U ( x ) [30, 34, 35]. These networks change—some interactions become moreprominent, while others fade as e.g., transcription factor activity changes in response to externalsignals [58–60]—and so do the quasi-potentials, taking the system through a qualitative changepoint.In this view (Figure 3) the landscape changes in response to signalling and the resulting minima ofthe quasipotential, U ( x ; ζ ), which now explicitly depends on an external signal or differentiationfactor, shifts position in statespace. In this view we make the stationary states of the systemdependent on ζ , that is x ∗ = x ∗ ( ζ )for the x ∗ that solve ∇ U ( x ∗ ; ζ ) = 0 . (18)The set of solutions, x ∗ , does not need to behave in a continuous manner with the control variable,as is clear, for example, if ζ induces a bifurcation. A priori it is not clear if the potential has tovary smoothly.For a transitory landscape, we could potentially treat the landscape associated with ζ as themacrostate, but we have to note here that the control variable, ζ , will in practice rarely be scalar:differentiation into a more specialised cell-type typically depends on more than a single molecularfactor [61]. From a merely conceptual tool the Waddington or epigenetic landscape has been slowly morphinginto a framework for the qualitative and quantitative analysis of real biological systems. Thereare two areas where further investigation and development are likely to bear fruit, and which wediscussed above.First, ideas from dynamical systems theory, Morse theory, catastrophe theory, and especiallyconcepts related to structural stability, have important implications for the mathematical analysisof dynamical systems.A crucial challenge to their widespread use is, however, that (i) they are typically restrictedto gradient systems; (ii) they are only valid for deterministic systems. There is some reasonto be hopeful that analysis of deterministic systems can be meaningful for our understandingof stochastic dynamical systems. But this may require detailed case-by-case analysis, as somehallmarks of deterministic dynamical systems, such as certain types of bifurcations, may nottranslate to their stochastic counterparts, or vice versa .The second point relates to applying statistical mechanics to cell differentiation. There is obviousappeal to doing so as has been detailed before. There are two shortcomings to this, however: (i)equilibrium statistical mechanics rests on assumptions that almost certainly do not hold in thecontext of living and changing systems; (ii) much of the appeal of statistical mechanics stems fromthe fact that entropic considerations can point towards the state a system will be in. Definingthe relevant macrostates is problematic; and translating empirical entropy estimates into e.g.,the likelihood of a given cell-state being obtained, is not possible in the conventional framework.There is, however, as we have sketched out here, some scope to resolve these outstanding issuesby adopting a non-equilibrium perspective, and better definitions of cellular macrostates.The concept of a transitory landscape [62], may be an attractive way of combining the dynamicalsystems perspective pioneered by Thom [28] and others [30, 63], with the statistical mechanicsperspective, especially if an appropriate non-equilibrium framework is used.0
Acknowledgments
We would like to thank the members of the
Theoretical Systems Biology Group for many fruitfuldiscussions of cell-fate decision making processes. Anissa Guillemin and Michael P.H. Stumpfgratefully acknowledge funding from the
Driving Research Momentum fund.
Conflict of interest
The authors declare that they have no conflict of interest.
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