An applied mathematician's perspective on Rosennean Complexity
AAn applied mathematician’s perspective on Rosennean Complexity
Ivo Siekmann a a Institute for Mathematical Stochastics, Georg-August-Universit¨at G¨ottingen,Goldschmidtstraße 7, 37077 G¨ottingen, Germany
Abstract
The theoretical biologist Robert Rosen developed a highly original approach for investigat-ing the question “What is life?”, the most fundamental problem of biology. Consideringthat Rosen made extensive use of mathematics it might seem surprising that his ideas haveonly rarely been implemented in mathematical models. On the one hand, Rosen propa-gates relational models that neglect underlying structural details of the components andfocus on relationships between the elements of a biological system, according to the motto“throw away the physics, keep the organisation”. Rosen’s strong rejection of mechanisticmodels that he implicitly associates with a strong form of reductionism might have de-terred mathematical modellers from adopting his ideas for their own work. On the otherhand Rosen’s presentation of his modelling framework, (
M, R ) systems, is highly abstractwhich makes it hard to appreciate how this approach could be applied to concrete biolog-ical problems. In this article, both the mathematics as well as those aspects of Rosen’swork are analysed that relate to his philosophical ideas. It is shown that Rosen’s relationalmodels are a particular type of mechanistic model with specific underlying assumptionsrather than a fundamentally different approach that excludes mechanistic models. Thestrengths and weaknesses of relational models are investigated by comparison with currentnetwork biology literature. Finally, it is argued that Rosen’s definition of life, “organismsare closed to efficient causation”, should be considered as a hypothesis to be tested andideas how this postulate could be implemented in mathematical models are presented.
Keywords:
Robert Rosen; Complexity; Network biology; mechanistic models; definitionof life ∗ Corresponding author
Email address: [email protected] (Ivo Siekmann)
Preprint submitted to Elsevier April 2, 2018 a r X i v : . [ q - b i o . O T ] A ug . Introduction When for the first time I heard about Robert Rosen’s life-long quest for the secrets oflife, his theory of (
M, R ) systems and his approach to complexity I didn’t quite know whatto make of all this. There was an obviously highly original idea for investigating a questionwhich is so hard to answer that it is, in fact, rarely asked: What is life? Also the methodsthat Rosen used for his work, borrowed from the highly abstract theory of categories, donot quite fit in the classical arsenal of the applied mathematician’s toolbox. Could categorytheory, an area of mathematics so abstract that, in fact, even some of its pioneers referredto it as “abstract nonsense” be successfully applied to a fundamental real-world question“What is life?” which at the same time happens to be one of the hardest scientific questionsthat one may possibly ask? That sounded interesting, very interesting, indeed!So I asked two questions that I usually ask myself when I hear about something newand exciting to me in science:1. Which of Rosen’s ideas can I steal for my own work? (more about stealing later, seeSection 5.3!)2. Do I believe Rosen’s answers to his research questions “What is life?” and “What isa complex system?”I will present my answers to these questions as my personal perspective on RobertRosen’s work. The purpose of this is two-fold: First, in my opinion, Rosen’s highly originalwork deserves more attention from the mainstream of mathematical biologists. Second, Ibelieve that Rosen’s frustration that his ideas were not more widely and openly accepted(Mikulecky, 2001) is not completely coincidental—there are important differences betweenRosen’s theoretical concept of a model and the understanding of modelling within theapplied mathematics community. These differences are on the one hand philosophical—Rosen demands that a model accurately represents the causal relationships between theelements of the system to be modelled (see Section 4) whereas models typically built by Rosen’s work on (
M, R ) systems is by no means the only application of category theory to the sciences.Best-known are perhaps applications in computer science— two examples for textbooks are Pierce (1991)and Barr and Wells (2012)—as well as mathematical physics (Coecke, 2011). A recent introduction tocategory theory with a view towards applications in the sciences by Spivak (2014) underlines the the factthat the trend of category-theoretic ideas in science is increasing. But Rosen’s work is one of the earliest,if not the earliest application of category theory outside mathematics. hypothesis regardinga possible mechanism underlying the system behaviour (see Section 5). On the other hand,Rosen applies mathematical notions, in particular, category theory, in a different spirit thanmost applied mathematicians would. This issue—which is related to Rosen’s presentationof his ideas rather than the ideas themselves—is more important than it may look at afirst glance because this difference in using mathematical tools may deter an audiencewith a mathematical background from Rosen’s ideas (Section 6.2.1). My presentation isbased on the original publications Rosen (1958a,b, 1959, 1971, 1973, 1991) and Rosen(2000) but I will most often refer to Rosen (1972) because this, in my opinion, is thebest summary of Rosen’s early publications and to his monograph “Life Itself” (Rosen,1991) which is the most comprehensive account of the philosophical basis of Rosen’s work.Another good introduction into Rosen’s thinking are his “Autobiographical Reminiscences”(Rosen, 2006).The article is structured as follows: In Section 2 we introduce the notions of metabolism-repair systems ((
M, R ) systems). In Section 3 we present Rosen’s proposed characterisationof life as systems that are “closed to efficient causation”. We show that this concept is not—as Rosen suggests—a specific property that can be deduced from the architecture of (
M, R )systems but should be regarded as a postulate, a hypothesis to be tested by implementing“closure to efficient causation” in mathematical models. Rosen’s specific view of modellingwhich is closely related to his interpretation of category theory is presented in Section 4. Idescribe the conceptual basis of mechanistic models in Section 5. In particular, I will arguethat Rosen’s relational models can be regarded as a specific type of mechanistic models.In the Discussion (Section 6) I compare mechanistic models with Rosen’s perspective onmodelling and present some ideas how his concept of an organism could be investigatedvia mathematical models in physiology and ecology.
2. Rosen’s answer to the question “What is life?”
Although most people—with or without a scientific background—seem to have a goodintuition when it comes to decide if something is “alive” it is nevertheless very hard tocome up with a rigorous scientific definition of life. Thus, definitions of life are usuallydescriptive—a list of properties that are characteristic of living systems is given such asthe following appearing in Campbell (2008):3.
Order.
Organisms are highly ordered, and other characteristics of life emerge fromthis complex organization.2.
Reproduction.
Organisms reproduce; life comes only from life (biogenesis) .3.
Growth and Development.
Heritable programs stored in DNA direct the species-specific pattern of growth and development.4.
Energy Utilization.
Organisms take in and transform energy to do work, includingthe maintenance of their ordered state.5.
Response to Environment.
Organisms respond to stimuli from their environment.6.
Homeostasis.
Organisms regulate their internal environment to maintain a steady-state, even in the face of a fluctuating external environment.7.
Evolutionary Adaptation.
Life evolves in response to interactions between organismsand their environment.But these properties are not necessarily defining: systems that are not usually consideredto be living systems may have one or even several of these properties. Indeed, Campbell(2008) refers to this list as emergent properties and processes of life rather than a definition.Instead of a descriptive definition, Rosen proposes a relational approach for distin-guishing systems that are “dead” from systems that are “alive”. He starts from a setof components that he explicitly refers to as black boxes i.e. he avoids making any as-sumptions on the internal structure of these components. Instead his focus is on therelationships between these components—he develops a highly abstract theory with thepurpose of demonstrating that the way that components interact determines if a systemis “complex” or “simple” and also, if a system is “alive” or “not alive”. By developing anapproach that intentionally ignores the properties of individual components of a systemand emphasising the relationships between these components he followed a motto of hismentor Nicolai Rashevsky (cited according to Rosen (2006))—“Throw away the physics,keep the organisation”.More generally, the question of the relationship between structure (i.e., for example,the underlying physics) and function in biology has a long history. For example, thefamous Cuvier-Geoffroy debate in front of the French Academy of Sciences in 1830 wasultimately about the two principles “form follows function” which was Georges Cuvier’sview whereas Geoffroy Saint-Hilaire argued for the opposite position “structure determinesfunction”. In Rosen (1991), his monograph “Life itself”, he strongly rejects “structure4etermines function” which is currently, for example, influential in molecular biology inthe theory of protein folding—because the sequence of amino acids (primary structure)to a great extent controls the three-dimensional arrangement (tertiary structure) and this3D structure determines the function of a protein it is argued that structure determinesfunction (Petsko and Ringe, 2008).In contrast, Rosen states that biological functions arise from the interactions betweenthe parts of a biological system, independent of the material realisation of the components.In order to explain this idea, let us consider calcium signalling. In many cases when hor-monal or electrical signals reach a cell, calcium oscillations are used for propagating thesesignals within the cell and control a wide range of cellular functions such as the contractionof heart cells or the transcription of particular genes. The shape of these oscillations canbe very different between cell types although the Ca signalling components involved arethe same—voltage-gated Ca channels, that allow calcium influx in response to electricalsignals, intracellular channels like the inositol-trisphosphate or the ryanodyne receptor,that release large amounts of calcium from intracellular stores when stimulated, and Ca pumps, that return Ca released to the cytosol back to intracellular stores. How can Ca oscillations be so different in different cell types if they are generated by similar sets of Ca signalling components? An obvious explanation is that Ca oscillations in particular celltypes are shaped by relationships between the components that are characteristic of thiscell type. This is the concept of the Ca “toolbox” which is the basis of our current un-derstanding of Ca oscillations (Berridge et al., 2000). But does the fact that differencesin the relationships between components are important for explaining the different shapeof Ca oscillations imply that we should restrict ourselves to investigating relationshipsand completely ignore structural properties of the components? We will come back to thisquestion in more general terms in the Discussion.From Rosen’s introduction of his ( M, R ) systems it is quite clear that he not only wishesto apply this approach for explaining the behaviour of particular biological systems butfrom the outset he aims for answering the grand question “What is life?”. As for manymodels, also for (
M, R ) systems the answer to his question “What is life?” is alreadydetermined to some extent by the construction of the model—this will be explained inmore detail in Section 5. Here, we will demonstrate that Rosen focuses mostly on twoof the aspects of life mentioned above, energy utilisation (in the following referred to asmetabolism) and homeostasis. 5he M in ( M, R ) systems refers to metabolism. Metabolism is formally modelled as thetransformation of “input materials” to “output materials” via the action of components .Mathematically, components are represented as mappings f : A → B, a (cid:55)→ b = f ( a ) (1)between sets A (“input materials”) and B (“output materials”). In biochemical terms, f may be interpreted as an enzyme because it catalyses the transformation of elementsof A to elements of B while remaining unchanged. However, in real metabolic networks,enzymes degrade so that f will eventually “disappear”. Thus, in order to ensure long-term stability, Rosen assumes for each component f of an ( M, R ) systems the existence ofa component Φ that “replicates” f should it be degraded. We denote such a component Φ f if we want to emphasise the fact that Φ f replicates the component f . The mappings Φ areagain components (called repair components ) as defined in (1)Φ f : C → H ( A, B ) , c (cid:55)→ Φ f ( c ) (2)but the range of Φ f must be H ( A, B ), the set of maps between A and B . Also, it is pos-tulated that the domain C of Φ f contains at least one environmental output O i.e. thereexits a subset of O ⊂ C that does not contain the domain of any component f (Rosen,1972). With the repair components Φ (the R in ( M, R ) systems), Rosen adds a repre-sentation of homeostasis to his (
M, R ) systems—each Φ f ensures the continuous operationof a particular component f . To give the repair components a biological interpretationsimilar to the enzymes f , Rosen sometimes refers to the Φ as genetic components. Thus, An adaptation of Rosen’s terminology to biochemistry may be found in Letelier et al. (2006). Here,we keep the original terminology. The nature of this “disappearance” is not made explicit in Rosen’s writings. He only states thatdisappearance of a component has the effect that the production of output material B stops. By using this notation, Rosen would like to imply that the maps appearing in H ( A, B ) are“(homo)morphisms”, maps that preserve mathematical structure associated with A and B rather thangeneral maps—see Section 4 for an explanation of the category-theoretic notion of morphism via an exam-ple. But because Rosen avoids assigning a specific mathematical structure to the sets A and B this hasno consequences for his model, in particular, the maps H ( A, B ) cannot model, for example, biochemicalproperties of metabolism. Moreover, in many circumstances the morphisms H ( A, B ) are still just a seteven if A and B have a particular mathematical structure. By introducing Φ f , strictly speaking, the set O fails to be an environmental output because it is nowcontained in the domain of Φ f . Unfortunately, whereas the degradation of components f can be prevented by the repaircomponents Φ, repairing these newly introduced components Φ would require another setof repair components. But Rosen was able to demonstrate that the infinite regress of havingto add more and more repair components could be avoided—he proposed that under certainassumptions a repair component for Φ f could be identified with an element from the rangeof f . Because a repair component for Φ f (which Rosen denotes β and names replicationmap ) therefore does not need to be added to the system, the infinite regress is avoided. Wewill discuss the construction of β in the next section, Rosen himself explains the detailsmost clearly from a a mathematical point of view in Rosen (1972).In summary, we observe that by suitably combining metabolic and repair components,an ( M, R ) systems is capable of achieving homeostasis. Although all components havea limited life time, the system is able to survive for much longer (in theory indefinitely)because components are replaced early enough before they degrade. Of course, this onlyrefers to the components that are parts of the (
M, R ) systems because at least some ofthem depend on environmental inputs. This demonstrates that (
M, R ) systems are able toautonomously maintain their internal organisation, provided that an “energy source” (viaenvironmental inputs) is available. We will explain Rosen’s own formulation of this resultin more detail in the next section.
3. Closure to efficient causation
In the previous section we explained that through interactions of metabolic and repaircomponents a (
M, R ) system achieves some level of autonomy—it is capable of maintainingits internal components (which all have a limited life time) by drawing on an energy sourcefrom the environment. Rosen summarised this this as “organisms are closed to efficientcausation”. In “Life Itself” he discusses in detail how his theoretical ideas relate to the In Rosen’s own biochemical interpretation of the components f as enzymes and Φ as genetic com-ponents, ( M, R ) systems can be related to the production of enzymes via gene transcription which itselfdepends on the activity of enzymes. But the abstract construction of (
M, R ) systems are general enoughthat the approach can be applied to different domains. one of which is the efficient cause . The efficient cause is mostclosely related to the modern notion of causality. In the context of ( M, R ) systems eachof the components are the efficient causes of the transformation of elements in the domainto elements in the range, for example, in (1), f is the efficient cause for transforming A to f ( A ) ⊂ B . For the same example, A provides the material cause for the transformationof A to f ( A ). We only consider the simplest (
M, R ) system, consisting of one metabolic component f and one repair component Φ f . In order to avoid degradation of the repair component Φ f we need an additional repair component β that replaces Φ f . Summarising (1), (2) andadding β we get A f −→ B Φ f −→ H ( A, B ) β −→ H ( B, H ( A, B )) . (3)Rosen develops a complicated argument why the map β in (3) can be identified with anelement b of the set B . He explicitly constructs a parameterisation B (cid:51) b (cid:55)→ β b : H ( A, B ) → H ( B, H ( A, B )) (4)that assigns a map β b to each b ∈ B . Rosen’s interpretation of (4) is that each β b is, in fact,already contained in set B and needs not be explicitly added to the system. Rosen’s con-struction of the β b has caused a considerable amount of confusion—some authors disputedits mathematical feasibility (Landauer and Bellman, 2002) whereas others responded tothis claim by explicitly constructing sets and maps as in (3) following Rosen’s approach(Letelier et al., 2006).In contrast, I will argue that from a mathematical point of view there is, in fact,nothing to show. The only restriction for β is the condition that β ( f ) = Φ f which meansin terms of ( M, R ) systems that β repairs Φ f by transforming f . After choosing any map β that fulfils this condition, a parameterisation β b as in (4) can be easily obtained—we only Applications of Aristotle’s classification of four causes have some tradition in biology, confer Tinber-gen’s levels of analysis, first presented in Tinbergen (1963). Rosen also discusses the formal cause and the final cause . Although all four causes are important forRosen’s theory, the remaining two Aristotelean causes are not directly relevant to our discussion. Thereforewe refer the reader to Rosen (1991). This is, in fact, not as strong a restriction as it may seem—by combining sets via Cartesian products ‘ × ’and defining maps on the products in an obvious way, several ( M, R ) systems with more than threecomponents can be cast in the form (3). b ∗ that maps to β b ∗ = β and map all other elements of B toarbitrary β b .Rosen’s difficulty arises because he insists on deriving the parameterisation β b fromevaluation maps, see Rosen (1972) where (cid:15) b is denoted ˆ b : (cid:15) b : H ( B, H ( A, B )) → H ( A, B ) , Φ (cid:55)→ Φ( b ) . (5)But there is no reason for constructing the maps β b in this way—the fact that the β b wereobtained from evaluation maps (cid:15) b (5) never plays a role in Rosen’s discussions of the repli-cation map β . In summary, as Rosen postulated, it is indeed possible to construct ( M, R )systems where all components are “repaired” by other components once they have sur-passed their finite life time. But the realisation of the map β does not, as Rosen suggests,follow from the architecture of ( M, R ) systems. Instead, I propose to consider the exis-tence of the replication map simply as a postulate regarding the structure of living systems,summarised in the statement “organisms are closed to efficient causation”.More interesting than the details of the mathematical investigation is Rosen’s inter-pretation of (
M, R ) systems whose elements interact in a way that mutually ensuresreplacement of failing components. (
M, R ) systems with this property provide Rosen’smodel for organisms which he characterises as “organisms are closed to efficient causa-tion”. In order to explain this concept, Rosen (1991, chapter 10) presents a diagram thatillustrates this statement (Fig. 1). The diagram shows—indicated by solid arrows—the“material” transformations between elements of the sets A , B , H ( A, B ), H ( B, H ( A, B ))and H ( H ( A, B ) , H ( B, H ( A, B )) (by a mapping to B ) but also shows “causal” relationships—indicated by broken arrows—where components initiate a transformation by acting on el-ements in one of these sets. By following a broken and then a solid arrow we see thatthe production of each component is “caused” by another component in the system—inRosen’s own words, f “entails” β b , β b “entails” Φ f and Φ f “entails” f . Thus, the systemcontains a closed loop of efficient causation f → β b → Φ f → f , a property that Rosendenotes closure to efficient causation .In summary, in this section we have explained Rosen’s proposed definition of livingsystems,“organisms are closed to efficient causation”. We have shown that “closure toefficient causation” is not a result that follows from the construction of ( M, R ) systems.In contrast, the ability of organisms to autonomously maintain their internal organisationshould be regarded as a postulate, a hypothesis to be tested for concrete biological sys-9 B Φ f fH ( A, B ) ∈ b ∈ β b (cid:55)→ H ( B, H ( A, B )) ∈ Figure 1:
Closure to efficient causation:
This diagram (adopted from Rosen (1991, chapter 10)) illustratesthe different processes that the individual components of the (
M, R ) system in (3) are involved in. On theone hand, solid arrows show where transformations from input material to output material occur— A istransformed to B , B is, in turn used for “repairing” f whereas f is transformed in order to repair Φ f . Onthe other hand, for the broken arrows, a component located at the start of an arrow indicates the initiationof a transformation located at the arrow tip— f catalyses the “metabolic” transition of A to B , Φ f actson B for repairing f and the replication map β b starts the repair of Φ f by transforming f . It is clear thateach of the components f , Φ f and β b regulates the repair of another component. In Rosen’s words, thesets A , B , H ( A, B ) and H ( B, H ( A, B )) can be regarded as “material causes” whereas f , Φ f and β b are“efficient causes”. Because each of the components is in turn produced by one of the other componentsthe system is closed to efficient causation . tems by developing mathematical models. We will return to this important idea in theDiscussion.
4. The modelling relation
Interesting about Rosen’s approach are not only (
M, R ) systems themselves but alsohow they are constructed and investigated. He uses category theory, one of the mostabstract mathematical disciplines, that was developed starting from the 1940s. In order togive a simple example that introduces many important aspects of category theory withoutrequiring much mathematical background consider a planar algebraic curve. Readersthat are familiar with category theory may safely skip this slightly lengthy example but I The following introduction to category theory is intentionally informal—the point is to introduce thespirit of category theory rather than enabling the reader to start their own career as a category theorist.See the Introduction of the classical monograph by Mac Lane (1971) for a very readable exposition for themathematically inclined reader. This is, in fact, the first example of category theory that I saw as a student. I thank Prof. HeinzSpindler (University of Osnabr¨uck, Germany) for his beautiful lectures on algebraic geometry that gaveme a lot of pleasure. modelling relation may be regarded as the representation of a functor between differentcategories. This has been explained more formally by Louie (2009).
An algebraic curve in the plane is defined by a polynomial equation in two indeter-minates x and y i.e. f is an element of a polynomial ring K [ x, y ]. Examples include theparabola, y − x = 0 or the unit circle, x + y = 1. Geometrically, the curve is the set ofpairs ( a, b ) that fulfil the equation f ( x, y ) = 0; the ( a, b ) are elements of a two-dimensionalvector space R . One of the most important aspects of modern algebraic geometry is notto be very specific about R , the interest is in general properties of algebraic sets definedby systems of polynomial equations. Whereas classical algebraic geometry investigatespolynomial equations over the complex numbers C , we may instead consider the real num-bers R , the rationals Q or a finite field with q elements F q . Even more generally, we maychoose an algebra R over a field K . A K -algebra is a vector space over a field K whoseelements can in addition be multiplied, unlike in a general vector space which only requiresaddition of elements. Because, trivially, a field can be regarded as a one-dimensional K -algebra over itself, this includes the examples of different fields given above. The generalitywith which R can be chosen is expressed quite clearly by the following category-theoreticdescription of an algebraic curve. For an arbitrary non-constant polynomial f ∈ K [ x, y ],the affine planar algebraic curve C over R with equation f is defined as the functor (cid:67) : (cid:65)(cid:108)(cid:103) K → (cid:83)(cid:101)(cid:116)(cid:115) (6)with (cid:67) ( R ) = { ( a, b ) ∈ R | f ( a, b ) = 0 } (7)for all K -algebras R . Thus, (cid:67) ( R ) is the set of zeroes of f over R which means that thefunctor (cid:67) provides us with a set for each K -algebra. From the category-theoretical point A field K is an algebraic structure where addition and multiplication of elements are associative , commutative and distributive and additive and multiplicative inverses exist. This means that for each a ∈ K we find a b ∈ K so that a + b = 0 ( b is denoted − a ) and for each c ∈ K , c (cid:54) = 0 there is a d ∈ K suchthat c · d = 1 ( d is denoted d − ). Examples are the complex numbers C , the real numbers R , the rationalnumbers Q or finite fields F q with q elements.
11f view, each K -algebra R is an object of the category of “all” K -algebras (cid:65)(cid:108)(cid:103) K and “all”sets are the objects of the category (cid:83)(cid:101)(cid:116)(cid:115) . But a category not only consists of objects butalso of maps between objects that preserve the algebraic structure of the objects, so-called morphisms . For (cid:83)(cid:101)(cid:116)(cid:115) the morphisms are just ordinary maps but for (cid:65)(cid:108)(cid:103) K these are K -algebra homomorphisms φ : R → S . As a map between categories, a functor not onlyrelates objects but also morphisms. For our example, for a K -algebra homomorphism φ : R → S between K -algebras R and S we obtain a corresponding map φ ∗ : (cid:67) ( R ) → (cid:67) ( S )between the sets (cid:67) ( R ) and (cid:67) ( S ) via φ ∗ : (cid:67) ( R ) → (cid:67) ( S ) , ( a, b ) (cid:55)→ ( φ ( a ) , φ ( b )) . (8)Because φ is a K -algebra homomorphism it is indeed true that if ( a, b ) is a zero of f (over R ), ( φ ( a ) , φ ( b )) is also a zero of f (over S ).This abstract representation of a planar algebraic curve as a relationship between K -algebras and the geometric objects of interest, the sets (cid:67) ( R ), has a punch line that maygive some insight why category-theoretical ideas have been quite successful in some areasof mathematics but, more importantly, clarify one of Rosen’s key ideas, the modellingrelation . It can be shown that for describing algebraic curves it is sufficient to consideronly one particular K -algebra, the coordinate ring A = K [ x, y ] / ( f ) , where ( f ) = f K [ x, y ] . (9)With the expression f K [ x, y ] we denote the set of all polynomials that contain the poly-nomial f as a factor. The crucial point is that the the coordinate ring A alone is sufficientfor finding the curves (cid:67) ( R ) for all K -algebras R . We remind the reader that the func-tor (cid:67) assigns the evaluation map φ ∗ : (cid:67) ( A ) → (cid:67) ( R ) defined in (8) to each K -algebrahomomorphism φ : A → R . It can be shown that we can calculate the curve (cid:67) ( R ) via φ ∗ ([ x ] , [ y ]) = ( φ ([ x ]) , φ ([ y ])) , [ x ] , [ y ] ∈ A. (10)by evaluating φ ∗ at the particular point ([ x ] , [ y ]). In category-theoretical terms, the coor-dinate ring A is called a representation of the functor (cid:67) via the universal element ([ x ] , [ y ]). Due to set-theoretic paradoxes, we can, in fact, not consider categories of “all” K -algebras or “all”sets. A (commutative) ring is an algebraic structure similar to a field. The difference is that multiplicativeinverses are not required to exist for all elements. Note that in addition to the ring structure the coordinatering also has the structure of a K -algebra. [ x ], [ y ] are equivalence classes and can be considered as those polynomials with “remainder” x or y , (cid:83)(cid:101)(cid:116)(cid:115) ,algebraic curves defined by f , can be understood by analysing a particular K -algebra, thecoordinate ring A . We will now—still with the example of a planar algebraic curve—explain that Rosen’sidea of a model can be understood as a functor between a natural and a formal system.Both the natural as well as the formal systems are represented as categories, in fact, shortlyafter the initial introduction (Rosen, 1958a), Rosen (1958b) redefined (
M, R ) systems usingcategory-theoretical ideas—we will discuss Rosen’s use of category theory in Section 6.2. Itis instructive to observe how the representation of the “planar algebraic curve functor” (cid:67) that assigns to a K -algebra R a curve (cid:67) ( R ) is used by mathematicians working in the fieldof algebraic geometry.Let us say that algebraic curves are the “natural system”, graphs obtained by findingthe zeroes of a polynomial equation f ( x, y ) = 0. In contrast, we regard K -algebras as a“formal system” that—according to Rosen—encodes the “causal entailments” present inthe natural system. In general, causal entailment refers to the relations between objectsdefined by the morphisms of a category which is quite abstract. But for the specific exampleof algebraic curves it is quite clear what this means. From a mathematical point of view,studying a general algebraic curve over a general K -algebra R is impossible without furtherassumptions on the K -algebra R because sets do not have a lot of structure. In contrast,for K -algebras, arithmetical operations such as addition and multiplication are defined andmathematical theorems have been proven that give insight into how the elements behaveunder these operations. Thus, whereas the structure of K -algebras may be investigatedwith a variety of tools from commutative algebra, much less insight can be gained bysimply considering the sets (cid:67) ( R ) of algebraic curves over R . But the functor (cid:67) enablesus to switch between the “natural system” (sets) and the “formal system” ( K -algebras)so that we can explore geometric facts using algebra. Incidentally, algebraic geometers arewell aware of this and refer to this process with the motto “Think geometrically, provealgebraically!” (Alekseevskij et al., 1991). respectively, when “dividing” by f —the coordinate ring is an example of a quotient ring . We will not gointo more detail because it does not add much to the discussion and refer the interested reader to anyintroduction to algebra. K -algebra R can be understood as a “model” of another K -algebra S because wecan “translate” the algebraic curve (cid:67) ( R ) over R to the algebraic curve (cid:67) ( S ) over S usingthe functor (cid:67) by assigning the evaluation map φ ∗ to each K -algebra homomorphism φ : R → S . Even better, because the functor (cid:67) has a representation via the coordinatering A (9) we may even resort to studying only one K -algebra, namely the coordinatering, and translate the results to any other K -algebra via evaluating the functor at theuniversal element (10).Whereas the functor (cid:67) provides us with an example of a model in Rosennean terms, wemay also look for an example of simulation in the same context. Rosen defined a simulationas a relationship that considers the natural system as a “black box” without attempting (orbeing able) to capture the “causal entailment” within the natural system. An example ofsimulation for our example is the numerical approximation of algebraic curves. Numericalmethods may succeed in obtaining an approximation of an algebraic curve without anyconsideration of the underlying algebraic structure by iteratively approximating points ofthe curve from a starting value ( x , y ) known to lie on the curve (Gomes et al., 2009).With Rosen we might say that these numerical methods are able to “predict” the “naturalsystem” i.e. the algebraic curve. But it is clear that this is not based on bringing theentailment structure of a formal system in congruence with the entailment structures ofthe natural system to be modelled. This, however, is according to Rosen the ideal that amodel should live up to. As I will explain in the Discussion, it is exactly this ambitionthat, in my opinion, separates Rosen’s understanding of modelling most strongly from anapplied mathematician’s view on modelling which is illustrated in the next section. An anonymous reviewer brought to my attention that two famous articles by Alan Turing providea good example for the difference between simulation and model. The imitation game (also known asthe Turing test) was proposed by Turing (1950) in order to answer the question “Can machines think?”.In order to pass the test the machine must communicate in natural language with a human evaluatorand through this conversation convince the evaluator that it is human. This is a perfect example forsimulation because by definition of the test it is unimportant if and to which extent the machine attemptsto accurately represent human intelligence. In contrast, Turing (1952) proposes a mathematical model thatexhibits inhomogeneous stationary distributions (Turing instability). This leads to a model that explainsmorphogenesis in terms of two interacting chemical species (an “activator” and an “inhibitor”) that diffusewith different speeds. . Lie, cheat and steal—the applied mathematician’s ways for finding the truth I believe that there is no elaborated philosophy of modelling in applied mathematicsthat could be compared to Rosen’s. So it might be helpful to first look at
DynamicModels in Biology , a highly readable introduction to mathematical biology by Ellner andGuckenheimer (2007). Near the end of the book the authors briefly introduce the “threecommandments of modelling”:1. Lie2. Cheat3. StealAfter explaining the three commandments in a bit more detail I will provide a de-scription of models in applied mathematics that will be compared with Rosen’s modellingrelationship in Section 6.
Everyone knows that models are based on assumptions. What not everyone knows isthat models (at least the good ones (Ellner and Guckenheimer, 2007)) are based on false assumptions. As illustrations we can take nearly all mathematical models from physics.One of the most striking example is the apparently harmless notion of a mass point.Moving bodies such as cars, space ships or parachutists are described by particles withoutspatial extension whose mass is concentrated in a single point. In this way our modeldescribes objects that may weigh several tonnes or more while at the same time they “arenot even there” because it is assumed to have no spatial extension! The only reason thatwe find such an idea not completely outrageous is by the justification we get for this modelafter we have applied it to a natural system. We “get away” with this obviously wrongassumption, we can, for example, predict the trajectories of celestial bodies to a certainaccuracy. Also, more detailed models of rigid body motion and the notion of the centre ofmass give additional support for this model and insight why representing bodies as masspoints worked in the first place. The most important point here is, though, that beforethis model was applied to a concrete problem, it was not at all clear that it would turn outto provide a useful description of a physical object. The justification of “lies” in modellingcan only be given in hindsight. 15 .2. Cheat
With cheating, Ellner and Guckenheimer (2007) mostly refer to a particular way ofusing statistics. They recommend to do things that “would make a statistician nervous” bystretching the limits within which statistical methods can be used instead of just “lettingthe data speak for itself”. It is not easy to provide an example for “cheating” becauseobviously scientists will usually not describe anything they did in a study as cheating.Because I would not like to accuse colleagues of cheating either I have no choice but togive an example of “cheating” from my own work. A few years ago I was working on amodel for an ion channel (Siekmann et al., 2012). My motivation was to take into accountmodel gating, a feature that is quite common in ion channel dynamics but which hasrarely been accounted for in models. Instead of continuously adjusting their activity manyion channels switch spontaneously between highly different types of behaviour (modes). Iwanted to demonstrate that across all experimental conditions each of the different modesdefined the same type of behaviour i.e. could be described by statistically similar models.Unfortunately, for some experimental conditions I was not able to fit a model to thesegments representative for the modes because the channel was switching too fast andtherefore the segments were too short in order to produce statistically conclusive results.But although I thus was not able to rigorously prove my claim I nevertheless argued thatthe hypothesis of modes which are unchanged across all experimental conditions was—withsome positive and in the absence of contrary evidence—presumably correct.This way of using statistical methods emphasises that experimental data is only one ofmany sources of knowledge that are synthesised in a mathematical model, thus, the fateof a model should not depend solely on the success or failure of a particular statisticalmethod.
Although it may sound even worse than lying and cheating, in scientific terms, stealingmight actually be the most acceptable of the three commandments. It simply means reusingideas of models that have previously appeared in the literature by, for example, applyingthem to new systems. A famous example is the well-known Lotka-Volterra model which canbe seen as the beginning of predator-prey modelling. Volterra (1926) simply reinterpretedthe law of mass action kinetics where the rate of a chemical reaction is assumed proportionalto the product of the concentrations of the two reactants as the catch rate of a predator16eeding on a prey. The reason that terms characteristic of chemical models were often“stolen” by ecologists and epidemiologists is because the law of mass action and enzymekinetics can, from a more abstract point of view, be interpreted as contact rates betweentwo populations (Siekmann, 2009).We briefly mention one danger of stealing—in comparison to the original domain ofapplication, a “stolen” model may increase the amount of lying and cheating—on theone hand the assumptions of the original model may be less valid and on the other handexperimental validation of the original model may not be available in the new context.
In order to compare the approach followed in typical models in applied mathematicswith Rosen’s modelling relation I will give a brief description of models in applied math-ematics. I will refer to these as “mechanistic” in the following because Rosen presumablymeans similar models when he refers to mechanistic models. However, I will argue in theDiscussion that at least some of his objections only arise when mechanistic models areinterpreted in a reductionistic sense.The aim of a mechanistic model is to provide insight into a natural system by synthe-sising different sources of knowledge. This is achieved by defining a formal system thattransparently captures how the elements of the system interact with each other and howthese interactions are parameterised with experimental data. A mechanistic model is alsostrongly determined by a purpose which means that already the architecture and not justthe interpretation of the model results is defined by the question that the model shouldanswer.Building a mechanistic model starts with the formulation of a set of assumptions thatsummarise what is known about a system, extended by some hypotheses regarding detailsof the natural system that are currently unknown. Which aspects of our knowledge arerepresented in how much detail depends on the model purpose. Based on the underlyingassumptions a model structure is constructed that is meant to represent the assumptions aswell as possible. This model can then be simulated in order to produce results. The resultsobtained from the model are then interpreted in comparison with the natural system andin the light of the assumptions made in the beginning which leads to conclusions that aredrawn from the modelling study.To give an example, I will refer to recent work in mathematical ecology that I con-17ributed to. A series of papers, starting with Bengfort et al. (2016a) and Siekmann andMalchow (2016), primarily had two purposes. The aim of Bengfort et al. (2016a) was toinvestigate alternatives to the classical model of population dispersal based on the commonmodel for diffusion due to Fick (Fick, 1855). Siekmann and Malchow (2016) consideredalternatives to the classical model for environmental fluctuations based on stochastic termsthat scale linearly in the population densities. Bengfort et al. (2016b) studied the combinedeffect of Fokker-Planck diffusion (Fokker, 1914; Planck, 1917), an alternative to Fickiandiffusion, and linear noise terms whereas Siekmann et al. (2017) combine Fokker-Planckdiffusion with a nonlinear noise term proposed in Siekmann and Malchow (2016). Con-sistent with the model purpose, the authors primarily consider a very simple model forthe interactions of populations, the Lotka-Volterra competition model. Also, a special pa-rameter set is considered where in the deterministic, non-spatial version of the model, thepopulation with the higher initial population always outcompetes its competitor. In orderto investigate spatial and stochastic effects, the parameter of the diffusion model and thenoise model are varied. The models demonstrate that stochastic fluctuations enhance thesuccess of invading species that invade the habitat of a resident population but may alsoenable resident and invader to coexist which is impossible in the deterministic non-spatialversion of the model.At this point it is important to note that the definition of a mechanistic model presentedhere by no means excludes relational models favoured by Rosen—in the terms developedhere, an important underlying assumption of a relational model simply is that the inter-nal structure of the individual system elements is not represented in detail. The mostimportant difference to Rosen’s interpretation of models is that no suggestion is madethat a mechanistic model accurately captures the causal relationships of the system to bemodelled—there is no modelling relationship that in a theoretical sense provides the mod-eller with access to the structure of the natural system. The reason for this is scepticism—alarge proportion of applied mathematicians would presumably be highly pessimistic thatachieving congruence between a formal system and a natural system as envisaged by Rosenwas possible and even if it could be achieved that this could be verified. Thus, from theoutset, the motivation of a mechanistic model is much more modest. The aim of modellingis to provide insight into an aspect of a natural system that is defined by the purpose ofthe model. Obtaining a complete understanding is, in principle, out of reach due to thesimplifying assumptions made when the model was built. Also, mechanistic models can18nly provide possible explanations of phenomena observed in the natural system. If theresults obtained from the model contradict the behaviour of the natural system one con-cludes that the underlying assumptions of the model are either incorrect or incomplete.But if the results are consistent with the system to be modelled we cannot conclude thatthe explanation provided by the model is correct because we cannot exclude the possibilitythat alternative models with completely different underlying assumptions produce similarresults. Instead of regarding a mechanistic model as a mathematical representation of some“truth” it is therefore more accurate to think of a model as an argument for a particular hypothesis explaining the observed behaviour of a natural system.Finally, it is widely accepted among applied mathematicians that not the developmentof individual models but the comparison of several competing models of the same systemthat are based on different assumptions provides most insight. Modelling is therefore notso far from studying a system via experiments—with the important advantage describedby the mathematician Vladimir Arnold with the words “mathematics is the natural sci-ence where experiments are cheap”. This is very well illustrated by several monographson mathematical biology for which we give a few examples—the general introductions byMurray (2002, 2003), Edelstein-Keshet (2005) and Ellner and Guckenheimer (2007) men-tioned above, also we refer to more specific books on ecology (Okubo and Levin, 2001;Malchow et al., 2008) or physiology (Keener and Sneyd, 2009a,b).
6. Discussion
As explained above, Robert Rosen looked for an answer to the question “What islife?” in a way that was different to the commonly used modelling approaches at histime. Following the motto “throw away the physics, keep the organisation” he proposed toinvestigate relationships between “components” without associating these abstract entitiesthemselves with any structure.Looking at this idea from the point of view of an applied mathematician, we observethat Rosen’s model starts with the “lie”, in the sense of Section 5.1, that the physicsof components that make up a biological organism is mostly irrelevant for understandingits functioning. A problem with this is that we can only learn in hindsight, once thismodel has been applied to a real biological organism, if this assumption has been able19o provide new insights. But because Rosen himself was more interested in developinghis formal framework and developing the theoretical ideas he drew from those studies, hispublications contain at most hints to possible applications.The idea of a relational approach to biology has, in the last two decades, becomequite influential under the name of “network science”—we refer to the recent textbook byBarab´asi (2016), one of the most influential figures of the discipline, as an introduction tothe large body of literature. Papers in network science often follow statistical approachesfor inferring networks from data and the resulting networks are analysed by computationaltechniques developed with methods from graph theory and statistical mechanics.Studies in the area of network science clearly illustrate the challenges with “throwingaway the physics and keeping the organisation”. For example, once a large-scale biologicalnetwork has been inferred from data, the question of its interpretation might not be easyto answer, precisely because only minimal assumptions are made regarding the propertiesof individual nodes. Is it most important that a network has certain global propertiessuch as being scale-free (Watts and Strogatz, 1998; Barab´asi and Albert, 1999), to whichextent the network can be controlled (Liu et al., 2011; Ruths and Ruths, 2014) or thatcertain “network motifs” are more prevalent than expected by chance (Milo et al., 2002;Alon, 2007)? If we consider, more specifically, gene regulatory networks, another problembecomes apparent. Rosen always assumes that relationships between the components ofhis (
M, R ) systems are known. Unfortunately, inferring the interactions within biochemicalnetworks such as the highly complicated large networks of genes and their transcripts isoften quite challenging and might not lead to conclusive results. This motivated, forexample, Oates et al. (2014) to include a more detailed model of the underlying reactionsin order to obtain more accurate results on the interactions between the components of thebiochemical network. Also, as far as the impact of the conclusions is concerned, one mightargue that combining network models with at least some description of the underlying“physics” of the components has been more promising than studies that are restricted tonetworks whose nodes without considering their underlying structure. As an example Irefer to Colizza et al. (2006) who investigated the global spread of epidemics along theairline traffic network using the example of the 2002 outbreak of severe acute respiratorysyndrome (SARS).In summary, by considering the example of “network science” which is the area ofscience that is probably most closely related to Rosen’s idea of a relational biology, the20enefit of Rosen’s proposal of throwing away the physics and keeping the organisationare not entirely clear. But this has to be considered in a situation where applications ofhis ideas are still in relatively early stages because Rosen himself did not work towardsapplying his theory to concrete biological problems.
As mentioned several times above, a strong view of Rosen’s is the motto “throw awaythe physics and keep the organisation”. For this reason he deliberately defines the com-ponents of his (
M, R ) systems as “black boxes”. Rather than describing the structure ofcomponents that form a system, his relational biology focuses on their interactions withother components. Consistently, he primarily uses category theory for describing interac-tions between objects; that the objects are members of categories with certain underlyingmathematical structures hardly plays any role.This use of category theory is likely to disappoint most readers of Rosen’s works witha mathematical background. In the preface of her recent textbook “Category Theory inContext” Emily Riehl introduces category theory like this (Riehl, 2016):Atiyah described mathematics as the “science of analogy.” In this vein,the purview of category theory is mathematical analogy . Category theory pro-vides a cross-disciplinary language for mathematics designed to delineate gen-eral phenomena, which enables the transfer of ideas from one area of study toanother.A strong motivation for the development of category theory and one of the main reasonsfor its success as a mathematical discipline is the formalisation of links between differentareas of mathematics. When category theory was originally formulated by Eilenberg andMacLane (1945) the new notions of category theory facilitated understanding the connec-tions between topological spaces and algebraic objects such as groups or vector spacesthat can be associated to them. The ability for finding such mathematical analogies (be-tween topological and algebraic objects in the case of algebraic topology) crucially dependsboth on relations between objects from particular categories (such as topological spaces Sir Michael Atiyah (*1929), British mathematician, one of the most distinguished mathematicians ofthe 20th century. Apart from many other awards and honours he won the Fields medal (1966) and theAbel Prize (2004).
21r groups) but also on the underlying mathematical structures of these objects that arepreserved by morphisms.
But, as already mentioned, Rosen explicitly avoids assigning structure to the compo-nents of his (
M, R ) systems. From the introduction of (Rosen, 1958b) it becomes clearthat Rosen regards category theory as an incremental extension of graph theory that en-ables him to more flexibly describe relations between “black boxes”. Unfortunately, thisprevents him from taking much advantage from the main strength of category theory,namely, relating the structure of mathematical objects appearing in different disciplines ofmathematics.The negative impact of this use of category theory on an audience of applied mathe-maticians must not be underestimated. A strong motivation in mathematics itself as wellas in the community of applied mathematicians is to use mathematical notions as effi-ciently as possible. By failing to take full advantage of the ability of category theory torelate mathematical structures Rosen does not only miss the chance to capture propertiesof a biological system that might be encoded in such structures. Even worse, an audiencefrom a mathematical background might even be deterred from Rosen’s ideas, not becauseof the ideas themselves but due to the perceived shortcomings in their mathematical pre-sentation. In summary, one might go as far as saying that instead of being a strength ofRosen’s theories, category theory is one of the most important obstacles for their accep-tance. Nevertheless one shall not be overly critical of Rosen’s approach of using categorytheory for his research—after all he was a pioneer in applying a novel, quite difficult math-ematical discipline at a time when even the foundations of this discipline were still underdevelopment.
Does the preceding section imply that category theory is not an appropriate tool forinvestigating biological systems? One reviewer of an earlier version of this manuscriptkindly directed me to work from the group of mathematical physicist John Baez who havemost recently applied category theory to open reaction networks (Baez and Pollard, 2017).It is worthwhile to compare this emerging research with (
M, R ) systems which were inspiredby networks of metabolic reactions. Rather than abstracting reaction networks to a networkof input-output relationships, Baez and co-workers go the opposite way—they develop22pecific categories
RNet and
RxNet in order to formalise Petri Nets, a diagrammaticrepresentation of chemical reactions. By constructing a functor to the category
Dynam ofopen dynamical systems they can relate a given Petri net to a system of ordinary differentialequations, the rate equations associated with this particular Petri net. A functor from
Dynam to the category of relations
Rel maps dynamical systems to their steady states.Of course, the ambition of Baez and co-workers is presumably not to answer questionslike “What is life?” but their “compositional framework” allows them to build reactionnetworks from simpler components via composition of morphisms and relate the structuralproperties of reaction networks to similar models such as electrical circuits (Baez and Fong,2016), signal-flow diagrams (Erbele, 2016) and Markov processes (Baez et al., 2015) whichare all formalised in a similar way as described above using the language of category theory.It seems clear that although this work is not less “relational” than Rosen’s (
M, R ) systemsor the network science approaches mentioned in Section 6.1, the publications from Baez’sgroup clearly take advantage of category theory for relating the different mathematicalstructures characteristic of different modelling approaches.
In “Life Itself”, Rosen (1991) repeatedly proposes relational models as alternativesto mechanistic models. But, first of all, according to the view explained in Section 5.4,mechanistic models should not be considered as the opposite of relational models, in fact,relational models can be regarded as a particular type of mechanistic model. Second,many of Rosen’s objections arise because he implicitly assumes that mechanistic modelsnecessarily have to be interpreted in a reductionistic way. But building a mechanistic modeldoes not mean that the natural system necessarily must be “reduced” to the mechanismrepresented by the model—in fact, this is just a specific interpretation. In contrast, inSection 5.4 we propose an alternative perspective on mechanistic models—according to thisview the “mechanism” represented in a mechanistic model only provides an explanation of aparticular aspect of the system behaviour which is defined by the underlying assumptions ofthe model. If additional aspects of the system behaviour are to be considered this requiresrefining the assumptions of the model, as a result the new model will represent a moredetailed mechanism that provides a more comprehensive (but still partial) explanation ofthe system behaviour. 23 .4. Rosennean complexity and mechanistic models
The difference between Rosen’s view on modelling and the view I outlined in Section 5.4is most obvious when considering his definition of complex systems:A system is simple if all its models are simulable. A system that is notsimple, and that accordingly must have at least one non-simulable model, iscomplex.Rosen’s concept of complexity is a direct consequence of his modelling relationship.With his modelling relationship he outlines an approach that enables us to directly relatea natural system to a formal system, the model. Thus, it might seem that modelling is, atleast conceptually, trivial. Indeed, for our example of an algebraic curve (cid:67) (Section 4.1),the representation of the functor (cid:67) by the coordinate ring A (9) provides us with a “model”for the algebraic curve over arbitrary K -algebras. But according to Rosen’s definition, thefact that the coordinate ring A exists as a model for arbitrary K -algebras makes algebraiccurves a “simple” system. For a “complex” system, an analogue of the coordinate ring A might still exist but it is “non-simulable” which Rosen defines as not Turing-computable.Although such a model would still perfectly describe the natural system, the requiredcalculations formalised by a Turing machine might not terminate in finite time.Most striking about Rosen’s definition of a complex system is how the relationship of asystem and “its” models is described. According to Section 5.4 and in contrast to Rosen’sconcept of the modelling relation a system does not “have” models—models cannot beobjectively associated with a system via the modelling relationship but rather are subjec-tively attributed to the system by the modeller. A model will—due to the requirementof making simplifying assumptions—necessarily always remain incomplete, the case of a“simple” system in Rosennean terms, where a perfect formal representation of a naturalsystem can be found, does not exist. Mechanistic models serve a specific purpose by effi-ciently representing a set of underlying assumptions that are consistent with the purposeof the model. As explained in Section 5.4, rather than being a formal representation ofa particular “truth” about a natural system, the aim of mechanistic models is to explorescientific hypotheses that are represented in the assumptions of the model. Therefore,consideration of competing models based on alternative assumptions is an important partof scientific discussion in the literature. 24 .5. Rosennean complexity in the toolbox of mathematical biologists The most important aspect of Rosen’s theories is his postulate that “organisms areclosed to efficient causation”. In order to to assess how well this notion is able to describeliving systems one has to go beyond theoretical considerations. From the perspectiveof a modeller it is therefore the most important shortcoming that so far there are veryfew examples of mathematical models implementing Rosen’s postulate in the context ofconcrete biological systems.In the field of modelling biochemical reactions, this issue is being addressed by the groupof C´ardenas and Cornish-Bowden. In a series of papers, Letelier et al. (2006); Cornish-Bowden et al. (2007); C´ardenas and Cornish-Bowden (2007) developed a simple (
M, R )system representing a simple biochemical reaction network that was later implemented ina mathematical model and investigated by simulation (Piedrafita et al., 2010, 2012a,b;Cornish-Bowden et al., 2013).Whereas organisms are postulated to be closed to efficient causation, they must beopen to “material causes” as explained in Section 3 or, in more familiar terms, flows ofmatter and energy. A mathematical model that realises Rosen’s concept of an organismshould therefore also appropriately account for the exchange of matter and energy with theenvironment. The bond graph methodology (Borutzky, 2010) is an established approachfrom the engineering literature for building models of complex systems with energy flowsbetween multiple domains (electrical, chemical etc.). Bond graphs were recently applied tobiochemical reactions (Gawthrop and Crampin, 2014; Gawthrop et al., 2015) and subse-quently to more complex physiological systems (Gawthrop and Crampin, 2016; Gawthropet al., 2017). Also, in mathematical ecology there are several examples of energy-basedmodels ranging from the early (
E, M ) framework developed by Smerage (1976) to themore recent studies by Cropp and Norbury (2012) and Bates et al. (2015). Extendingthese frameworks by modelling approaches that realise closure to efficient causation willenable us to investigate the significance of Rosennean Complexity for ecosystems.In all implementations of Rosen’s principle of closure to efficient causation an obviousdifficulty is to identify “efficient causes” and distinguish them from “material causes”—Rosen’s own publications provide relatively little guidance due to the mostly formal presen-tation with very few specific biological examples. In this regard, the recent paper Mossioet al. (2016) is highly relevant: the authors develop a theory of biological organisationthat comprises Rosen’s views but draws from the much longer tradition of organicism.25rganicism is a perspective on biology which states that “organisms are the main objectof biological science because [...] they cannot be reduced to more fundamental biologicalentities (such as the genes or other inert components of the organism)” (Mossio et al.,2016). Organicism implies that the individual parts that the organism consists of can onlybe understood by taking into account their relationships and interactions with other parts.But crucially, in contrast to the approach that Rosen takes with his (
M, R ) systems, Mossioet al. (2016) avoid reducing the parts of an organism to “black boxes” without underlyingstructure. Instead, they identify specific parts of biological systems as constraints whichcontrol processes without themselves being altered by them—they give the role that en-zymes play in reaction networks and the influence of the vascular system on the flow ofoxygen in the body as examples. Biological organisation according to Mossio et al. (2016)is realised via “closure of constraints” which shows that constraints are a closely relatedconcept to Rosen’s “efficient causes”. This idea of biological organisation is one of threetheoretical principles for biology proposed in the highly readable special issue “From thecentury of the genome to the century of the organism: New theoretical approaches” pub-lished in
Progress in Biophysics and Molecular Biology —the others are variation (Mont´evilet al., 2016) and a postulated biological “default state” (Soto et al., 2016). The articlesfrom this special issue provide valuable guidance to modellers who wish to construct modelswhich represent Rosen’s idea of an organism or, indeed, stand in the much longer traditionof organicism.Although it seems clear that the task of building such mathematical models that providea better representation of organisms will not be an easy one, it seems equally clear that thiswill bring us another step further towards answering the grand question “What is Life?”.
Acknowledgements
I cordially thank two anonymous reviewers for their thoughtful and supportive reviewswhich greatly improved this article. I am particularly grateful to both reviewers for direct-ing me to highly relevant additional literature.
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