The information theory of individuality

Despite the near universal assumption of individuality in biology, there is little agreement about what individuals are and few rigorous quantitative methods for their identification. Here, we propose that individuals are aggregates that preserve a measure of temporal integrity, i.e., “propagate” information from their past into their futures. We formalize this idea using information theory and graphical models. This mathematical formulation yields three principled and distinct forms of individuality—an organismal, a colonial, and a driven form—each of which varies in the degree of environmental dependence and inherited information. This approach can be thought of as a Gestalt approach to evolution where selection makes figure-ground (agent–environment) distinctions using suitable information-theoretic lenses. A benefit of the approach is that it expands the scope of allowable individuals to include adaptive aggregations in systems that are multi-scale, highly distributed, and do not necessarily have physical boundaries such as cell walls or clonal somatic tissue. Such individuals might be visible to selection but hard to detect by observers without suitable measurement principles. The information theory of individuality allows for the identification of individuals at all levels of organization from molecular to cultural and provides a basis for testing assumptions about the natural scales of a system and argues for the importance of uncertainty reduction through coarse-graining in adaptive systems.

From the perspective of physics and chemistry, biological life is surprising. There is no physical or chemical theory from which we can predict biology, and yet if we break down any biological system into its elementary constituents, there is no chemistry or physics remaining unaccounted for (Gell-Mann 1995). The fact that physics and chemistry are universal—ongoing in stars, solar systems, and galaxies—whereas to the best of our knowledge biology is exclusively a property of earth, supports the view that life is emergent. This stands in contrast to the universality of chemical phenomena which can be predicted from quantum mechanical considerations in fundamental physics even when this proves to be computationally cumbersome or intractable (Defranceschi and Le Bris 2000). The asymmetry in what can be gleaned from working down toward ever more elementary constituents versus working up through levels of aggregation is captured by the terms reductionism and emergence (Anderson 1972; Laughlin and Pines 2000). It is often difficult to predict physical properties of aggregates from knowledge of constituents, and this extends to questions of behavior where it is rarely clear how far “down” to go (Anderson 1972; Krakauer and Flack 2010a; Flack 2017b). There are assumed to be dominant microscopic scales for a given set of aggregate properties yet our understanding of what constitutes a fundamental unit (Gilbert et al. 2012; Daniels et al. 2016) and whether these units count as individuals, have implications for many areas of science, from taxonomy and cladistics through to physiology, behavior, and ecology (Clarke 2011; Wilson and Barker 2013).

It is almost inconceivable for us to imagine a biological science without a concept of units or individuality. After all, how could we speak about metabolism, behavior or the genome without first establishing a unit or container of observation and measurement? Even Schrödinger in his prescient book, What is Life? (Schrodinger 2012), sought to explore the persistence of biological phenotypes of organisms—or even features of ecosystems—through the lens of elementary and universal physical underpinnings, made strong prior assumptions about the reality of individual organisms:

“What degree of permanence do we encounter in hereditary properties and what must we therefore attribute to the material structures which carry them? The answer to this can really be given without any special investigation. The mere fact that we speak of hereditary properties indicates that we recognize the permanence to be of the almost absolute. For we must not forget that what is passed on by the parent to the child is not just this or that peculiarity...Such features we may conveniently select for studying the laws of heredity. But actually it is the whole (four- dimensional) pattern of the phenotype, all the visible and manifest nature of the individual, which is reproduced without appreciable change for generations, permanent within centuries—though not within tens of thousands of years—and borne at each transmission by the material in a structure of the nuclei of the two cells which unite to form the fertilized egg cell. That is a marvel.”

Schrödinger did not set out to derive the individual from fundamental physics but to reconcile existing and rather traditional conceptions of individuality (essentially the individual as synonymous with the observable organism) with the new physics of quantum mechanics.

In this respect, Schrödinger was adopting a typically reductionist perspective, explaining features of biological science through first principles of physics (Weinberg 1995). In Schrödinger’s case, the physical feature of greatest importance to biology was the long-lived covalent bond. But for many reasons this line of approach has failed to deliver the deep and unifying insights based on physics (Anderson 1972), from which powerful biological ideas—such as adaptation or individuality—might be derived (Dupré 2009; Keller 2009).

The question we seek to address is more limited. How do we identify individuals without relying on features like cell membranes that may be solutions to challenges faced by particular systems for maintaining integrity rather than foundational properties? We want to allow for the possibility that microbes and loosely bound ecological assemblages such as microbial mats and cultural and technological systems, when viewed with a mathematical lens, qualify as individuals even though their boundaries are more fluid than the organisms we typically allow. It may also be the case that entities currently considered individuals are indeed individuals but not in the way we think—organisms are more complicated than typical individuality definitions acknowledge. Humans for example contain approximately as many self-cells as symbiotic microbes (Andreu-Moreno and Sanjuán 2018), yet until recently with the advent of the concept of “holobiont” (Gilbert et al. 2012), the microbe portion of the human cellular ecosystem was not typically considered part of the human individual.

In an ideal case, visitors to an exoplanet would have a procedure for identifying or “perceiving” individuals based on a quantitative survey with minimal prior knowledge of the type of life form that they expect to encounter. In the next sections of the paper, we briefly review a few key standard assumptions about individuality in biology and challenges to formalizing the concept. We then discuss a way forward and develop an information-theoretic formalism.

Here, we briefly review some of the criteria currently used to identify individuals. For a synoptic treatment of individuality definitions see (Clarke 2011; Gilbert et al. 2012). A standard assumption is that replication presupposes individuality (Wilson and Barker 2013). Under this assumption, replicators typically include organisms that have developed from a fertilized egg, with individuality residing at the phenotypic level (see Dawkins 1983), and asexual microbes or clonal organisms for which individuality is defined based on shared genetic ancestry (Hughes 1990). The replicator assumption has served as the starting point for theorizing about what an individual is in a broad class of studies and out of this work has come three additional widely accepted properties of biological individuals: (1) they can increase in relative frequency by exploiting a source of metabolic free energy, (2) they respond adaptively to their environments, and (3) they are characterized by tightly coordinated relationships (chemical, physiological, computational) among their parts. The association of these properties with individuality has raised debate about whether individuality applies only to “single” organisms, as the replicator assumption suggestions, or also to cells and aggregates like societies (Gilbert et al. 2012).

Beyond replicators as proxies for an individuals, almost all definitions of individuality assume a set members (individual) and a set complement (environment). These are articulated in different ways including: (1) as an immunological concept pertaining to the idea of self and non-self (Pradeu 2012), (2) as a temporal aggregate encoding a common past separable or independent from the past of other aggregates (ontogenetic or phylogenetic) (Rieppel 2013) (3) as a spatially bounded collection of metabolic reactions insulated by a membrane from reactions in the environment (Rasmussen et al. 2004), and more abstractly, (4) as a unit of selection and evolutionary change (Buss 1987; Hughes 1990; Szathmáry and Smith 1997; Callcott and Sterelny 2011).

To reveal limitations of the above definitions and other hidden assumptions, a useful exercise is to consider aggregates and processes that do not typically get classified as individuals (Santelices 1999).

Work on social insects and on a number of plant, fungal and prokaryotic species demonstrates the possibility of individuality simultaneously at multiple organizational levels—physically distinct ants form aggregations called colonies and these colonies may be divided into spatially noncontiguous subsets (Gow et al. 2008; Esser et al. 2001). Furthermore, in many ant species the majority of worker ants do not replicate and the colony as whole does not replicate, but contiguity between past and future is nonetheless a feature of the system. And, importantly, it is the combination of reproduction by a minority of colony members coupled to the industry of the majority that allows the colony as a whole to adapt in response to changes in the environment. Taken together, these two observations suggest it is possible to have individuality without replication and some forms of individuality benefit when replication is partial.

Viruses occupy a figurative twilight zone in biology. Declared by some non-living, and treated by most as a rather pathetic minimal limit of life, viruses constitute obligate translational parasites, incapable of completing their life cycles without first appropriating the protein synthesis machinery of a host cell. The viral capsid contains a largely inert genome responsible for encoding only a small fraction of the proteins required for synthesizing a new virus genome and the capsid required for egressing from the infected cell. The virus exists only within the larger dynamical, regulatory network of the cell. Hence, the virus—understood as the active parasitic agent—is comprised largely of host encoded factors. And yet it can replicate, adapt, and has a persistent identity that distinguishes it from its “host” environment—despite the fact it relies on its “host” environment for replicating. And, recent work suggests that viruses like microbes form collective units that facilitate infection (Andreu-Moreno and Sanjuán 2018). These observations suggest that viruses in aggregate are individuals but not in the conventional sense. Rather they are what Krakauer (Krakauer and Zanotto 2006) has called “chimerical individuals.”

The above examples are fascinating but without a rigorous definition of both the environment and the agent it is difficult to speak consistently of individuals. This is analogous to figure-ground separation in gestalt psychology or computer vision. The background of an image carries as much if not more information than the object, and the challenge is to separate the two rather than assume that they are already distinct and independent.

One possibility is that ant colonies and viruses [or humans, for that matter, composed of 37 trillion microbes and 30 trillion “human” cells (Andreu-Moreno and Sanjuán 2018; Gilbert et al. 2012)] are only nominally individual—a categorization resulting from human perceptual bias for certain kinds of aggregation. But if they are real in a deeper physical sense then how might we determine this? We propose:Individuality can be continuous, with the possible surprising result that some processes possess greater individuality than others.Individuality can emerge at any level of organization. This requires we dispense with privileging a single level or object—for example, replicating cells or organisms—and then defining individuality based on features of these objects, such as sequestered germ cells, vertical transmission of genetic material, a common pool of metabolic free energy, or coordinated immune responses. Although these features may indeed be effective proxies when we have significant prior knowledge of a system, our goal should be to find fundamental, rather that derivative, properties of individuality. Defining individuality around derived properties risks precluding the possibility of individuality in at super-organismal levels and in distributed systems.Individuality can be nested. Given that life is hierarchically organized into trophic and functional levels, we allow the possibility of multiple, parallel levels of individuality.

We take this position to be related to the recent suggestion of (Rieppel 2013) where he argues for individuals based on hierarchical complexes of homeostatic properties and (Flack 2017a) who has proposed biological systems are information hierarchies resulting from the collective effects of components estimating, in evolutionary or ecological time, regularities in their environments by coarse-graining or compressing time series data and using these perceived regularities to tune strategies. As coarse-grained (slow) variables become for components better predictors than microscopic behavior (which fluctuates), and component estimates of these variables converge, new levels of organization consolidate.Allowing individuality to be continuous rather than binary, nested, and possible at any level, opens the door for more quantitative takes on familiar open questions in evolutionary theory including the relation between the units of selection and temporal and spatial correlation and whether individuality at one scale impacts coherence and autonomy and “lower” and “higher” scales. These revealed time and space scales and their interdependencies should provide clues about the mechanisms driving their consolidation and through this consolidation the emergence of individuals (Flack 2017a, b).

Given our proposition that individuals are aggregates that “propagate” information from the past to the future and have temporal integrity, and that individuality is a matter of degree, can be nested, distributed and possible at any level, how can we formalize individuality?

We will take as our starting point measurements from a stochastic process. This could be a vector of chemical concentrations over time, the abundance of various cell types, or probabilities of observing coherent behaviors. We use coarse-grained or quantized information-theoretic filters the quantize the measurements. Some of these filters will reveal a coordinated pattern of behavior, whereas others will filter out all signal and detect nothing. Thus, signal amplitude given an appropriate filter becomes a means of discovering different forms of individuality. This is somewhat analogous to observing patterns in infrared that would be invisible using the wavelengths of visible light—individuality is revealed through characteristic patterns of information flow.

The basis for this approach to aggregation comes from information theory, and throughout this paper we assume that individuals are best thought of in terms of dynamical processes and not as stationary objects that leave information-theoretic traces. In this respect, our approach might reasonably be framed through the lens of “process philosophy” (Rescher 2007) which makes the elucidation of the dynamical and coupled properties of natural phenomena the primary explanatory challenge. From the perspective of “process philosophy,” the tendency of starting with objects and then listing their properties—“substance metaphysics”—places the cart before the horse.

Our proposal that individuals are aggregates that propagate information from the past to the future and have temporal integrity can be viewed as a pragmatic operational definition that captures the idea there is something persistent about individuals. However, our motivation for defining individuality this way is actually much deeper. It lies in the information-theoretic interpretation of entropy, its connection to the physical theory of thermodynamics, and formal definition of work introduced by Clausius in the 1860s [see (Müller 2007) for an introduction to this history].

Briefly, work (displacement of a physical system) is produced by transferring thermal energy from one body to another (heat). Entropy captures, or measures, the loss in temperature over the range of motion of the working body. In other words, entropy measures the energy lost from the total available energy available for performing work. The insights of Clausius were formalized and placed in a mathematical framework by Gibbs in 1876.

In 1877, Boltzmann provided in his kinetic theory of gasses an alternative interpretation of entropy. For Boltzmann entropy is a measure of the potential disorder in a system. This definition shifts the emphasis from energy dissipated through work to the number of unobservable configurations (microstates) of a system, e.g., particle velocities consistent with an observable measurement (macrostate), e.g., temperature. The thermodynamic and Boltzmann definitions are closely related as Boltzmann entropy increases following the loss of energy available for work attendant upon the collision of particles in motion during heat flow. There are many different microscopic configurations of individual particles compatible with the same macroscopic measurement, and only a few of which are useful.

In 1948, encouraged by John von Neumann, Claude Shannon used the thermodynamical term entropy to capture the information capacity of a communication channel. A string of a given length (macrostate) is compatible with a large number of different sequences of symbols (microstates). A target word will be disordered during transmission in proportion to the noise in a channel. If there were no noise, each and every microstate could be resolved and the entropy would define an upper limit on the number of signals that could be transmitted. The study of the maximum number of states that can be transmitted from one point to another across a channel, in the face of noise and when efficiently encoded, is called information theory.

Shannon did not describe entropy in terms of heat flow and work but in terms of information shared through a channel transmitted from a signaler to a receiver. The power of information theory derives in part from the incredible generality of Shannon’s scheme. The signaler can be a phone in Madison and the receiver a phone in Madrid, or the signaler can be a parent and the receiver its offspring. For phones, the channel is a fiber-optic cable and the signal pulses of light. For organisms the channel is the germ line and the signal the sequence of DNA or RNA polynucleotides in the genome. Increasing entropy for a phone-call corresponds to the loss or disruption of light-pulses, whereas increasing entropy during inheritance corresponds to mutation or developmental noise. The same scheme can be applied to development, in which case the signaler is an organism in the past and the receiver the same organism in the future. One way in which we might identify individuals is to check to see whether we are dealing with the same aggregation at time t and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+1$$\end{document}t+1. If the information transmitted forward in time is close to maximal, we take that as evidence for individuality.

In its simplest form, Shannon made use of the following formal measures when defining information. The entropy H of a random variable S measures the uncertainty or information of the states that it can adopt:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H(S) = -\sum _i P(s_i)\log _2 P(s_i) \end{aligned}$$\end{document}H(S)=-∑iP(si)log2P(si)where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_i$$\end{document}si are the possible values of the state and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(s_i)$$\end{document}P(si) the probabilities of these states. For a coin there would be two possible values for S, heads and tails, and the values of these states for a fair coin would be the probability 0.5, yielding a metric entropy value of 1. Deviation from a fair coin corresponds to a reduction in information, as in the limit of bias where only one side of the coin is favored, the outcome is known in advance and any toss of the coin is perfectly predictable. This produces an entropy value of 0. Hence, information is minimized when predictability is maximized.

To capture the communication value of information Shannon introduced a signaler–receiver structure, which is now typically described using two random variables S and R. The maximum information transmitted between signaler and receiver is given by the Mutual Information (I). The I can be written in several different forms. One intuitive expression is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S;R) = H(S) + H(R) - H(S,R) \end{aligned}$$\end{document}I(S;R)=H(S)+H(R)-H(S,R)where H(S) and H(R) are the entropies of the signals, and H(S; R) the joint entropy of the two variables,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H(S;R) = -\sum _i \sum _j P(s_i, r_j)\log _2 P(s_i, r_j) \end{aligned}$$\end{document}H(S;R)=-∑i∑jP(si,rj)log2P(si,rj)The joint entropy is at a maximum when there is no relationship between the S and R variables. The I is therefore high when the information in S and R are high and they are strongly coupled in their values (H(S; R) is low). The I measures the information shared between S and R over a communication channel, because the only source of structure in R is assumed to come from S.

Another conventional way of writing I is,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S;R) = H(R) - H(R|S) \end{aligned}$$\end{document}I(S;R)=H(R)-H(R|S)where H(S|R) is the conditional entropy of R or the amount of information in R that is not in S. Hence, if all the information in R comes from S then H(R|S) will be zero, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S;R) = H(R)$$\end{document}I(S;R)=H(R).

If one of the random variables, for example the sender S consists of two parts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\{S_1,S_2\}$$\end{document}S={S1,S2}, we can decompose the mutual information using the chain rule (Cover and Thomas 1991)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_1,S_2;\,R)=I(S_1;\,R)+I(S_2;\,R|S_1) \end{aligned}$$\end{document}I(S1,S2;R)=I(S1;R)+I(S2;R|S1)with the second term being the conditional mutual information\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_2;\,R|S_1):=H(R|S_1)-H(R|S_1,S_2). \end{aligned}$$\end{document}I(S2;R|S1):=H(R|S1)-H(R|S1,S2).These measures provide the necessary statistics for an informational theory of the individual.Fig. 1The causal diagram of the system–environment interaction

When we model the interaction between a system and its environment we have to consider a more complicated situation which involves two channels. To be more precise, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {S}}}$$\end{document}S and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}$$\end{document}E be the state set of the system and the environment. For simplicity, we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {S}}}$$\end{document}S and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}$$\end{document}E are finite. The dynamics of the system is influenced by its own state, but it can also be influenced by the state of the environment.

This can be modeled in terms of a channel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi : {{\mathcal {E}}} \times {{\mathcal {S}}} \rightarrow {{\mathcal {S}}}$$\end{document}φ:E×S→S, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (e,s;s')$$\end{document}φ(e,s;s′) denotes the probability of the next system state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s'$$\end{document}s′ given that the current system state is s and the environment is in state e.