**Abstract**

Many biological phenomena do not have adequate mathematical representations. This is because living systems deploy logic and semiotics beyond our modern conceptions of mathematics and computation. Current approaches to complex problems rely on modelling, but one aspect of the problem is that they rely on a single form of mathematics, switching from it to another to address the next aspect, and so on. All this switching is an indication of how inadequate our mathematical tools are to date (ODE/PDE systems, stochastic models, discrete state-transition systems, topological algebra, etc.). Biological systems function at all these levels simultaneously, so why can’t mathematics do the same? It is not biology that is too messy to be modeled; it is our *use *of current mathematical paradigms that are not able to adequately address these biological problems. I claim that it is impossible to make significant progress in theoretical biology and personalized medicine without a breakthrough paradigm change towards *biology-driven *– not “biology-inspired”(!) – mathematics and computation. My goal is to accelerate scientific development in life sciences through *profoundly new theoretical foundations. *What I am pursuing is a unified view and understanding of life through a new perspective at mathematics and computation that goes *over scales *and is capable to incorporate also* informal and phenomenological aspects of information *about it.

**Research Scope**

Today, scientists use a multiplicity of mathematical and computational approaches, all based on the classical pattern of Turing machine models. There are three main toolkits for theoretical research in biology and medicine at he moment:

- quantitative modeling: ODE/PDE systems, probabilistic/statistical/stochastic (Monte Carlo, Markovian, Bayesian, etc.) models;
- qualitative modeling: discrete state-transition systems such as Boolean/logical networks, Petri nets, process algebras (CCS, CSP, ACP, LOTOS), etc.;
- visualization of experimental/empirical evidence: regulatory charts/maps/graphs, 3D VR models, simulation and animation, etc.

Combining two or more of these is called ‘hybrid modeling’. We distinguish between *hypothesis*-driven (qualitative[1]) and the *data*-driven (quantitative) approaches, both using visualization methods based on the combination of computer graphics/animation and data analytics to test their models, yet with (almost) no links between them. The majority of theoretical research in biology done up until now has been on modeling complex molecular networks. A large body of work has been done in *data*-driven modeling using systems of differential equations (Olmholt, 2013) and in *hypothesis*-driven approaches based on discrete state-transition systems, e.g. by using Petri nets (Heiner and Gilbert, 2013). However, progress in these areas has been limited, partly because we know little about the reaction constraints (initial and boundary conditions) and partly because the nature of most interactions between the participating components (genes, proteins, cells, etc.) has yet to be elucidated.

In most theoretical biology models the following statements hold true:

- data-driven and hypothesis-driven models are decoupled;
- all space-scales of interest are non-integrated;
- either upward or downward causation is used alone (generally upward, i.e. bottom-up models);
- no self-organization or criticality properties are addressed;
- no phenomenological (first person) descriptions are considered.

In particular, the *data driven approach *in life sciences has not lived up to its promise of predictive diagnostics and personalized medical treatment yet, (Ahn et al., 2006; Gomez-Ramirez and Wu, 2012). The extensive knowledge of the genome sequences of human beings and various pathogenic agents has led to the identification of only a limited number of new drug targets (Drews, 2003).

The employment of new methods for drug discovery based on strategies like high-throughput screening, combinatorial chemistry, genomics, proteomics and bioinformatics does not currently yield the expected new medications and therapies. A number of biotechnological projects such as gene therapy, stem-cell research, antisense technology and cancer vaccination have failed to deliver the expected results (Kubinyi, 2003; Glassman and Sun, 2004). The common problem with many of these innovative techniques, as was the case for gene therapy (Williams and Baum, 2003), is that the risks and unwelcome side effects have been underestimated. We still don’t know well the complexity of the process interdependences in biomolecular networks. On the other side, systematic histological analysis when investigating the pathology of tumors is very difficult. The problem is in providing exact and robust classification of the tissue samples from biopsy slides regarding their morphological properties, which determine the character/type of the formation, its degree of malignity/benignity, i.e. development stage and ultimately the patient’s survival chances. Modern virtual microscopy deploys sophisticated geometrical measurements for 3D model reconstruction, statistical comparisons with historical records in large databases and machine learning methods to deliver reliable automation diagnostics. However, the precision of such automated data analyses is still insufficient at large, because of the characteristic manifestations of the individual parameters (size, structure, texture, etc.), impeding the tumor categorization. Briefly, we deal with a remarkable spatial and temporal heterogeneity of the cancer cell assemblies, which does not provide any reliable clues about the nature of the underlying processes. The links between the specifics of the multicellular morphology and the subcellular biochemical processes, incl. substrate/biomarker/ion concentrations are sill missing.

A major problem in quantitative modeling is the assumption that all physics, and hence biology, can be calculated using iterative methods applied to differential equations (usually second order PDEs). But Norbert Wiener’s Cybernetics showed that systems with feedback require *integro-differential equations* for their elucidation, which introduces immense complications when trying to solve them. Also hierarchy theory (Salthe, 2012) suggests that *reductionism can never explain how novel properties and processes emerge.* Biological entities have properties that differ from chemical and physical ones and demand the invention of appropriate mathematical abstractions. Furthermore, a recent study (Bard et al., 2013) has put forward two essential principles of modern biology:

*No level of preferred status exists,*with necessary and important events taking place at and between each level so that causality occurs upwards, downwards and within levels (Noble, 2012; 2013). This rule has been shown for being realized within the entire physiome (genome, proteome, interactome) (Werner, 2005; Shapiro, 2013); we assume it holds also for the phenome.*Feedback between the participants at several levels*– with*oscillations being inherent*in the dynamics of this feedback in order to guarantee stability[2]. This rule defines the ‘first law of system biology’: Any complex biological event involves activity at many levels and the properties that emerge from this activity are not necessarily predictable (Kolodkin et al., 2012).

A source of current methodological problems in system and molecular biology lies in a preoccupation with the quantitative methods and a disregard for the qualitative, structural approach (Schroeder, 2013). Theoretical biology has been focusing largely on dynamics, stochastic processes and discrete mathematics. Nevertheless, neither graphical pathway maps, nor “ODEs or formal programs can indicate whether a description is complete in the sense that it mirrors the full complexity of reality with … its alternative pathways. Even if it predicts a correct observation, we are not to know whether the description is correct; we merely know that it is a possible solution. It is not obvious that there is any way of handling the depths of biological complexity…” (Bard et al., 2013). Also, the explosion of combinatorial complexity when dealing with vast numbers of variables and parameters in large dynamic multi-protein complexes leads to the insight that conventional modeling approaches fail to describe living systems processes. *The ease of data collection has exacerbated the need of models rich enough to explain relations present in the data.* But model selection must be perceived as an integral part of data analysis (Johnson and Omland, 2004). Besides, the researchers face *incomplete and noisy data *and the modeling environments have *no embedded intelligence*. Such an example is self-assembly (Tschernyschkow et al., 2013). A hierarchical levels approach could help here. *But any attempt to deal with multi-scales using conventional techniques is doomed to failure no matter how much data we collect.*

The problem is with their interpretation and that one cannot go between adjacent scales in a real existing hierarchy by using the logic of either of the scales – the intermediate region is multiply fractal (Cottam et al., 2004)! Local solutions require global knowledge (Cottam et al., 2013), but mathematics and not data remains the key to biology (Justman, 2015).

Now the time has come to rethink the foundations of biological and medical discovery. My research is intended to overcome these shortcomings. I believe in a *Fifth Paradigm* lurking behind the currently embraced Fourth Paradigm of Big Data in science, (Hey et al., 2009; Herman, 2013; Chu, 2014). This is the paradigm of *Integration*, which fully complies with the Integral Biomathics approach. To adequately address a specific issue, such as the one of multi-scales in biology, we may need to apply multiple mathematical approaches (qualitative and quantitative, invent new ones and link them together using a tailored *scheme. *Here *fractional calculus* (Oldham and Spanier, 1974) and *fractal geometry* (Mandelbrot, 1983) appear to be appropriate. But we need in biology far more expressive ways for integrating local ‘logics’. Besides, since logic is based on (natural) language used in human communication, new methods to transcend the limitations imposed by language and generalize logic beyond it into* a semantically rich logic of information *suitable for expressing complex systems and life were suggested in (Schroeder, 2011).

Another recent study shows that a system at criticality constitutes a perfectly self-observing entity by reducing its own quantum wave packets and providing the basis for *a completely new kind of information, different from digital information* (Hankey, 2015). The key properties of these new information states include: i) an internal loop of *phenome information flow*, accounting for the sense of ‘self’, for Husserl’s internal sense of time passing, and for Heidegger’s ‘being in time’, and ii) high, long-range coherence, revealing the new information states as constituents of an *integrated information theory* *of living systems* based on gestalts. Such states are supposed to support all aspects of ‘internalization’ in phenomenological studies and the cognizance of *experience*. The above novel approaches to logic and information in living systems come indeed very close to the biosemiotics stance (Kull, 2015; Brier, 2015). *Exploring the virtues of higher order (bio)logics and criticality state information could lead towards explaining the emergence of the multi-level hierarchy of life.* Stimulating the development of such advanced techniques to *naturalize computation* is precisely what characterizes my research interests. The community I have been organizing around this scope since 2011 consists now of *28 distinguished research fellows, 56 senior group leaders* and presumably around 10.000 sympathizers[3] from 47 countries worldwide. We recognize the necessity for going beyond traditional forms of quantitative methods, which provide only an *illusion of precision*, but do not serve well in the analysis of structural or otherwise ‘qualitative’ characteristics of reality, in particular of living systems exhibiting structural complexity (Salthe, 2008) well beyond what is investigated in physical sciences.

What I am interested in is to understand Nature, as René Thom argued in defense of his notion of *qualitative mathematics* (Thom, 1977). I claim that we need no more detailed physical models of biological systems that could handle ever larger and larger amounts of data from increasingly fine-grained studies of biological systems, but ways of identifying the biological properties that are as unique to such complex aggregations as temperature is to a set of molecules (Root-Bernstein, 2012). However, whereas classical physics discusses the one-, two- and many-body problems, complex networks with multiple feedbacks (Simeonov, 2002; Ehresmann and Simeonov, 2012) introduce *new emerging phenomena*, in particular with *non-linear criticalities*, and *interconnected time structures,* as essential characteristics of living systems. Therefore I reckon that we need *a new biomathematics* – a mathematics based on such fields as category theory and algebraic geometry which treats a number of mathematical abstractions (continuous and discrete functions, sets, groups, fields, rings, maps, etc.) within a unique *evolutionary*, *scalable* and *integrative* (ESI) framework*:* a mathematical and computational platform that deals with the emergence and development of biological organization from non-random selections amongst replicating variations of complex populations of entities. In fact, such biologically relevant mathematical concepts providing a base for realizing the preferred hypothesis-driven scenario *do *currently exist. Reflecting work by (Root-Bernstein, 2012), the above questions were raised with reference to a *‘classical’* view of the different domains of mathematics.

This view has been modified with the introduction of more unified and ‘relational’ approaches, such as category theory (Rosen, 1958a/b; Mac Lane, 1998; Ehresmann and Vanbremeersch, 2007) and algebraic geometry (Felix et al., 2008), as noted by (Hoffman, 2013), or spatial computing based on algebraic topology (Giavitto and Spicher, 2008). In addition, richer Bayesian and contextual models (Gomez-Ramirez and Sanz, 2013; Kitto and Kortschak, 2013), but also quantum models (Gabora et al.; 2013), post-Turing computation (Siegelmann, 2013) and elaborate forms of logic (Goranson et al, 2013; 2015; Marchal, 2013) have changed our understanding about the role of mathematics in biology. These approaches are part of our *new* framework for theoretical biology: *Integral Biomathics* (Simeonov, 2010a/b; Simeonov et al., 2011; Simeonov et al. 2012a/b; Ehresmann and Simeonov, 2012; Simeonov et al., 2013; Simeonov, 2015; Simeonov and Cottam, 2015).

The authors cited in the last two paragraphs are members of our worldwide research community. Our aim is not to develop new models. Nor is it to advance bioinformatics or biocomputation, understood as cellular/molecular/biochemical/DNA computing. Our goal is to support the development of relevant strategies, concepts, techniques and methods *underpinning the foundations* of an innovative mathematics and computation for biology, targeting crucial challenges in life sciences and medicine such as *internalism* (Matsuno, 1996; 2013; Cazalis, 2013) that could not be tackled satisfactorily with computational system biology to date. For instance, I am currently interested in answering the question how such an innovative mathematics and computation can relate and interpret the results in both NGS and the digital pathology of cancer. Tracing the non-locality of emergence, which is probably deeply related to the phenomena known in modern physics, is another issue of special interest for me.

**Research Focus**

I am interested in studying the temporal aspects of *emergence* and *robustness *in living systems, in particular those related to information storage and transmission in the following research context.

** Emergence.** Emergent properties in terms of multiple negative feedback and feed-forward controls lead to dynamical characteristics that cannot be adequately predicted by using linear mathematical models that disregard cooperation, competition, and non-additive effects. In my focus are models of emergence both in an explicit and in an implicit way, i.e. (a) by developing new formalisms and (b) by using simulations, without any explicit reference to the emerging properties[4]. Emergence theory regards the natural world as organized in terms of hierarchies (Pattee, 1973; Salthe and Matsuno, 1995; Kim 1999, Morowitz, 2002; Salthe, 2012; Matsuno, 2015) and heterarchies (McCulloch, 1945; von Goldammer et al., 2003; Cottam et al., 2005; Gunji et al., 2008; Sasai and Gunji, 2008) that have evolved

*over time*. Whereas reductionists defend an ‘upward causation’ by which molecular states bring about the higher-layer phenomena, the proponents of emergence advocate a ‘downward causation’ by which the higher-layer systems influence lower-layer ones. Indeed, there are three versions of

*downward causation*: weak, moderate and strong (Andresen et al., 2000). I adopt a combination of the moderate accent of downward causation (Bickhard, 2004) requiring a

*process ontology*and the “inside-out” causation concept

__of__Sydney Brenner (Brenner, 2010) in the context it is represented by Denis Noble (Noble, 2006; 2012; 2013). However, this requires the elaboration of further details with respect to the boundary constraints (incl. non-holonomic ones), in particular when making morphogenetic decisions. The only way to anticipate emergence in the Waddington-Thom-Salthe sense, and in particular with respect to ‘natural biocomputation’ (Salthe, 2013) I am interested in, is to observe and trace the pattern formations and their relations

*within their own context at possibly multiple layers*below, at and above the layer of their initial emergence and beyond (Cardinale and Arkin, 2012), including non-locality.

** Robustness.** Living systems tend to be impervious to less than drastic changes in their environment because they are able to adapt and have

*flexibly redundant*components and pathways that can act as backups if individual components or paths fail (Csete and Doyle, 2002; Kitano, 2002). The

*flexible redundancy*is at the base of the MES’ Multiplicity Principle (Ehresmann and Vanbremeersch, 2007). It means, so in Edelman’s “degeneracy” (Edelman & Gally, 2001), that the same output can be generated by structurally and functionally different components and pathways. The latter are a priori

*hidden*for the researcher due to switching off particular known components or paths (e.g. an enzyme that triggers the synthesis of an undesired protein).

This principle is a new key insight for understanding such phenomena as cancerogenesis and MODS[5]. It needs thorough consideration when developing future personalized medical therapies. Of course, the *complex functional interdependence* among the above characteristics of living systems needs further investigation.

The objective of a *biology-driven mathematics* pursued by Integral Biomathics is an appraisal of *second-order logic *in place of the first-order logic with which most standard mathematicians are familiar. This kind of “implicate order” (Bohm, 1980) should be also capable of capturing *acategorical elements*, which cannot be registered systematically, but only *systatically*[6]. *The great advantage of second-order logic over first-order is within its systatic capacity.* One drawback of second-order logic is, however, that it is not decidable. Because of this weakness, most standard mathematicians still like to adhere to first-order logic, which is decidable or provable. How to make it decidable is a big issue for biology (and physics). In order to specify meaningful statements framed in second-order logic, some sort of *qualifier* needs to be used. Information, as a natural phenomenon of a layered, filtered and shared content (syntax, semantics, semiotics) is a crucial factor for making the second-order logic, that is inevitably alone on the theoretical ground, decidable as revealed in the origins of life. The central point here is that *the phenomenology of information is intrinsically decidable if the internalist stance *(Matsuno, 1996; 2013), the one of the first person observer-participant (Vrobel, 2015), *is adopted*. The Integral Biomathics approaches targeting this aspect are phenomenology-based (e.g. Simeonov, 2015; Matsuno, 2015; Hankey, 2015; Hipólito, 2015; Ehresmann & Gomez-Ramirez, 2015) and bio/cyber-semiotics-based (Peirce, 1869; Kauffman, 2001; Kull, 2015; Brier; 2015; Nakajima, 2015). I am naturally attracted to both of them.

My research focus comprises *Second-Order and Higher-Order BioLogics.* Currently I am pursuing the investigation of specific problems characteristic of cancer research, neuroscience, and the origin of life, which are addressed within the following two research areas (Simeonov et al., 2013; Simeonov and Cottam, 2015):

- a
*hidden morphology theory*about the linkage between natural selection and form-function interactions targeting effectively both the optimization of forms to carry out particular functions, and the evolvement of novel functions from existing forms. This involves the development of appropriate geometrical projection and visualization tools that can easily model*processes*such as the complex folding of proteins or chromosomes, or detailed embryological development. Mathematical forms share little with the actual biological processes that give rise to such structures. The mathematical abstractions currently used in system biological models generally do not illuminate the processes that give rise to biological geometries, but only their outward forms, despite the seminal work of René Thom (1977). What is interesting about biological forms is not their topology per se, but the ways in which these forms are*reifications of the biochemical and biomechanical processes*which they carry out or make possible. We know that the folding of chromosomes is a prerequisite to bringing together genes that would otherwise be spatially separated, and also that*spatial proximity*permits the rapid diffusion and control of interactive gene products that would otherwise be unable to interact in a reasonable biological time frame across an unfolded genome (Junier et al., 2010; Wilczynski and Furlong, 2010). Human developmental biology has now excellent data concerning the sets of genes that must be turned on and when they must be activated or inactivated in order to produce proper embryological development. However, the*discrete**information*generated from combinations of individual genes is expressed as a*continuous flow*of proteins and hormones that produce gradients, which must be reified as organized groupings of cells that have a specific form. Therefore, embryology is also stymied by the lack of mathematical approaches that can link discrete, continuous and geometrical information simultaneously. But what kind of mathematical notions would make it possible to model simultaneously the effects of spatial structure on continuous functions such as diffusion that in turn regulate on-off gene regulatory switches that act discontinuously or digitally? Lewontin has stressed the*reciprocal relationships between genes, organisms and their environment*,*in which all three elements act simultaneously as both causes and effects*(Lewontin, 2002), a logic incommensurable with present day computational logic. In addition, modeling circularity and recursion in living systems (Kauffman, 2015; Simeonov and Cottam, 2015) and Turing Oracle Machines (Turing, 1939) are challenging problems I am interested in. *a realistic fractalization theory*: The resolution of the dichotomy between the one-person and third-person description of systems requires a subtle account of the interaction between multiple-time scales to rigorously describe complex internal (first person) experiences (Rössler, 1987;1998; Matsuno 1989; 1996) modeled externally (from a third person’s perspective) as hierarchical nested events (Vrobel, 2013) including the phenomenology of time itself (Vrobel, 2015; Simeonov, 2015).

These research areas are going to play a paramount role when shaping and progressing the new kind of biological mathematics and computation we are after in Integral Biomathics.

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[1] We adopt this simple convention although some hypotheses can be also quantitative, such as the square-root law in mathematics, derived from theoretical ideas

[2] Of course, non-oscilatory systems are also stable.

[3] The Springer statistics reports 4895, 2460 and 661 chapter downloads of (Simeonov et al., 2012a) for 2014, 2013 and 2012 correspondingly, while ResearchGate.net − 153 downloads of (Simeonov, 2010) and 454 downloads of (Simeonov et al., 2012b) and Academia.edu 405 downloads in total with the source countries. Accessed: 13.09.2015.

[4] For instance the shape of a galaxy can be derived by simulation (most models apply Newton’s law or variants thereof) over a large number of “particle” stars. The shape of a galaxy then is implicit in Newton’s law.

[5] Multiple Organ Dysfunction Syndrome

[6] when related to combination or synthesis (systatics)