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Non-algorithmic Mathematical Models of Biological Organization
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Non-algorithmic Mathematical Models of Biological Organization

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E3: Mathematical Biology".

Deadline for manuscript submissions: 30 September 2025 | Viewed by 8991

Special Issue Editor

Dr. Sergio Rubin

E-Mail Website
Guest Editor
Department of Communication, Art and Media, IULM University, 20143 Milan, Italy
Interests: morphogenesis, mathematics of open systems; category theory; halting problem; structural stability; catastrophe theory

Special Issue Information

Dear Colleagues, 

Understanding the biological organization from non-algorithmic mathematical models is one of the most important fields in mathematical biology. However, biological thinking in this century witnessed the overwhelming use of the algorithmic and syntactic mathematics, that cannot deal with the autopoietic self-referential organization of living systems. Algorithmic simulations not only misplaced the mathematical foundations of biology and its diverse areas such as genetics, cell biology, immunology, neurobiology, ecology, evolution, earth system and cognitive science, but also fostered the computational tool as the only method to encoding biological causality.

What is evident, even from the most basic process at the molecular level, is that algorithmic and computational biology is limited to explain processes and forms of living system such as, for example, the ‘protein-folding problem’. Therefore, if difficulties are already encountered at this level, it is to be expected that cellular, ecological or cognitive processes are even more distant from having algorithmic referents.

Therefore, in order to address the multiple scales and realizations of living systems autopoietic organization, it is necessary to develop or extend the comprehension of non-algorithmic mathematical models such as the (M,R)-system that on the one hand incorporate the right expression of causality for biological organization and on the other hand leverage both the increasing modelling power of category theory, calculus of indications or algebraic biology.

We welcome articles, reviews, communications, hypothesis, essays and opinions that aim to advance the use of the (M,R)-system and other non-algorithmic mathematical models in the understanding of the causality of biological systems organization involving self-fabrication -autopoiesis- and hence cognition, autonomy and anticipation and the fundamental differences with respect of artificial systems and non-biological systems. We welcome articles related to any of these topics:

  • Categorical and indicational models of the causality of biological organization;
  • Modelling relation between autopoietic and (M,R)-systems;
  • Formal and causal differences of biological and non-biological systems;
  • Cognition, autonomy, anticipation and self-reference in the causal organization of biological systems;
  • Scales of realization or instantiation of autopoietic and (M,R)-systems;
  • Novel non-algorithmic mathematical methods to study biological organization.

Dr. Sergio Rubin
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • non-algorithmic mathematical models
  • causal organization
  • biological and non-biological systems
  • (M,R)-systems
  • autopoiesis
  • category theory
  • calculus of indications, algebraic biology, self-reference

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Published Papers (3 papers)

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Research

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29 pages, 716 KiB  
Article
Using Relational Biology with Loop Analysis to Study the North Atlantic Biological Carbon Pump in a ‘Hybrid’ Non-Algorithmic Manner
by Patricia A. Lane
Mathematics 2024, 12(24), 3972; https://doi.org/10.3390/math12243972 - 18 Dec 2024
Cited by 1 | Viewed by 907
Abstract
Biologists, philosophers, and mathematicians building upon Robert Rosen’s non-algorithmic theories of life using Relational Biology and Category Theory have continued to develop his theory and modeling approaches. There has been general agreement that the impredicative, self-referential, and complex nature of living systems negates [...] Read more.
Biologists, philosophers, and mathematicians building upon Robert Rosen’s non-algorithmic theories of life using Relational Biology and Category Theory have continued to develop his theory and modeling approaches. There has been general agreement that the impredicative, self-referential, and complex nature of living systems negates an algorithmic approach. Rosen’s main goal was to answer, “What is Life?”. Many believe he provided the best but minimum answer using a cellular, metabolism–repair or (M, R)-system as a category-theoretic model. It has been challenging, however, to incorporate his theory to develop a fully non-algorithmic methodology that retains the essence of his thinking while creating more operational models of living systems that can be used to explore other facets of life and answer different questions. Living systems do more than the minimum in the real world beyond the confines of definition alone. For example, ecologists ask how living systems inherently mitigate existential risk from climate change and biodiversity loss through their complex self-organization. Loop Analysis, a signed graph technique, is discussed as a hybrid algorithmic/non-algorithmic methodology in Relational Biology. This methodology can be used at the ecosystem level with standard non-algorithmic field data as per McAllister’s description of the algorithmic incompressibility of empirical data of this type. An example is described showing how the North Atlantic Carbon Pump, an important planetary life support system, is situated in the plankton community and functions as a mutualistic ecosystem chimera. It captures carbon from the atmosphere as an extended (M, R)-system and processes it until it is sequestered in the marine sediments. This is an important process to alleviate climate change in magnitude equal to or larger than the sequestration of carbon on land with forests. It is suggested that the ecosystem level should replace the cellular and organismic levels as the main system unit in biology and evolution since all life exists and evolves with full functional potential in ecosystem networks and not laboratory test tubes. The plankton ecosystem is the largest after the total biosphere and consists of evolutionary links and relationships that have existed for eons of time. If there was ever a genuine robust, highly self-organized ecosystem, it would be planktonic. Severing the links in these thermodynamically open networks by focusing on lower levels of the biological hierarchy loses the critical organization of how life exists on this planet. There is no theory to regain this crucial ‘omitted’ ecological relational causality at the cell or organismal levels. At the end of the paper, some future directions are outlined. Full article
(This article belongs to the Special Issue Non-algorithmic Mathematical Models of Biological Organization)
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Review

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29 pages, 343 KiB  
Review
Robert Rosen’s Relational Biology Theory and His Emphasis on Non-Algorithmic Approaches to Living Systems
by Patricia A. Lane
Mathematics 2024, 12(22), 3529; https://doi.org/10.3390/math12223529 - 12 Nov 2024
Cited by 1 | Viewed by 1855
Abstract
This paper examines the use of algorithms and non-algorithmic models in mathematics and science, especially in biology, during the past century by summarizing the gradual development of a conceptual rationale for non-algorithmic models in biology. First, beginning a century ago, mathematicians found it [...] Read more.
This paper examines the use of algorithms and non-algorithmic models in mathematics and science, especially in biology, during the past century by summarizing the gradual development of a conceptual rationale for non-algorithmic models in biology. First, beginning a century ago, mathematicians found it impossible to constrain mathematics in an algorithmic straitjacket via öö’s Incompleteness Theorems, so how would it be possible in biology? By the 1930s, biology was resolutely imitating classical physics, with biologists enforcing a reductionist agenda to expunge function, purpose, teleology, and vitalism from biology. Interestingly, physicists and mathematicians often understood better than biologists that mathematical representations of living systems required different approaches than those of dead matter. Nicolas Rashevsky, the Father of Mathematical Biology, and Robert Rosen, his student, pointed out that the complex systems of life cannot be reduced to machines or mechanisms as per the Newtonian paradigm. Robert Rosen concluded that living systems are not amenable to algorithmic models that are primarily syntactical. Life requires semantics for its description. Rashevsky and Rosen pioneered Relational Biology, initially using Graph Theory to model living systems. Later, Rosen created a metabolic–repair model (M, R)-system using Category Theory to encode the basic entailments of life itself. Although reductionism still dominates in current biology, several subsequent authors have built upon the Rashevsky–Rosen intellectual foundation and have explained, extended, and explored its ramifications. Algorithmic formulations have become increasingly inadequate for investigating and modeling living systems. Biology is shifting from a science of simple systems to complex ones. This transition will only be successful once mathematics fully depicts what it means to be alive. This paper is a call to mathematicians from biologists asking for help in doing this. Full article
(This article belongs to the Special Issue Non-algorithmic Mathematical Models of Biological Organization)

Other

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9 pages, 299 KiB  
Opinion
Robert Rosen’s Anticipatory Systems Theory: The Science of Life and Mind
by Judith Rosen
Mathematics 2022, 10(22), 4172; https://doi.org/10.3390/math10224172 - 8 Nov 2022
Cited by 3 | Viewed by 4820
Abstract
When I am at conferences, talking about the scientific work of my father (theoretical biologist Robert Rosen, 1934–1998), I am often asked which aspects of his work I think are most important. My answer is Anticipatory Systems Theory. It’s about the entailment and [...] Read more.
When I am at conferences, talking about the scientific work of my father (theoretical biologist Robert Rosen, 1934–1998), I am often asked which aspects of his work I think are most important. My answer is Anticipatory Systems Theory. It’s about the entailment and characterization of both life and mind. It explains the fundamental nature of all life, showing how the human mind is an evolutionary concentration of the same peculiar behavior patterns manifested by all living organisms, regardless of species. How can we hope to fully understand ourselves or anything else in the biosphere of Earth without an accurate scientific comprehension of the entailment patterns underlying and generating all of it? The physics of orbital mechanics or atomic particles is insufficient for this. Therefore, I spend a lot of my time working to make the meaning of my father’s scientific discoveries accessible to as many human minds as possible. I think humanity is going to need this work in the future, and already needs it now. This paper will examine the basic premises of Anticipatory Systems Theory and describe, using examples familiar to all of us from daily life, how we can recognize Anticipation at work in ourselves and in local ecosystems all over the planet. I will conclude with some important ramifications of this theory, including how Anticipation necessarily plays into evolutionary processes. I will also point out the vulnerabilities of Anticipatory Systems (i.e., living organisms) to rapid change in environment, potentially leading to extinction cascades. Full article
(This article belongs to the Special Issue Non-algorithmic Mathematical Models of Biological Organization)
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