Systems Biology: Biological Networks, Network Motifs, Switches and Oscillators

Systems Biology: Biological Networks, Network Motifs, Switches and Oscillators

 

 

From Biological switches and clocks

The living cell receives signals from its environment and its own internal state, processes the information, and initiates appropriate responses in terms of changes in gene expression, cell movement, and cell growth or death. Like a digital computer, information processing within cells is carried out by a complex network of switches and oscillators, but instead of being fabricated from silicon transistors and quartz crystals, the cell’s computer is an evolved network of interacting genes and proteins. In the same way that computer design was made possible by a sophisticated theory of electronic circuitry, a basic understanding of cellular regulatory mechanisms will require a relevant theory of biomolecular circuitry. Although the ‘engineering mindset’ is sorely needed to make sense of the cell’s circuitry, the squishy, sloppy, massively parallel, analogue nature of biochemistry is so different from the solid-state, precise, sequential, digital nature of computers that the mathematical tools and intellectual biases of the solid-state physicist/electrical engineer are not entirely appropriate to unravelling the molecular logic of cell physiology. New modelling paradigms and software tools are evolving to meet the challenges of the new ‘systems biology’ of the living cell.

 

 

System Biology includes study of the following among other areas.

  • Biological Networks
  • Network Motifs
  • Switches
  • Oscillators

 

 

Biological Networks

  • Protein–protein interaction networks
  • Gene regulatory networks (DNA–protein interaction networks)
  • Gene co-expression networks (transcript–transcript association networks)
  • Metabolic networks
  • Signaling networks
  • Neuronal networks
  • Between-species interaction networks
  • Within-species interaction networks

 

Network Motifs:

  • Coherent Feedforward Loop (FFL)
  • Incoherent Feedforward Loop
  • Feedback Loop
  • Scaffold Motifs
  • Bi Fan
  • Multi Input Motifs (MIM)
  • Regulator Chains
  • Bi-Parallel
  • Single Input Module (SIM)
  • Dense Overlapping Regulon (DOR)

 

Biological Switches

  • Ultrasensitivity
  • Switches (Bistability)

 

Biological Oscillators

  • Clocks
  • Negative Feedback Only Oscillators
    • Repressilator
    • Pentilator
    • Goodwin Oscillator
    • Frazilator
    • Metabolator
  • Negative + Positive Feedback Oscillators
    • Meyer and Strayer model of Calcium Oscillations
    • van der Pol Oscillator
    • Fitzhugh-Nagumo Oscillator
    • Cyanobacteria Circadian Oscillator
  • Negative + Negative Feedback Oscillator
  • Negative and Positive + Negative Feedback cell cycle Oscillator
  • Fussenegger Oscillators
  • Smolen Oscillator
  • Amplified Negative Feedback Oscillators
  • Variable link Oscillators

 

Synthetic Biology study design of networks, switches, and oscillators.

 

From The dynamics and robustness of Network Motifs in transcription networks

Network Motifs

Even though biological systems are extremely complex, some of its complexity could be simplified. The study of a complex system in its entirety could prove impossible with current theories and technology. However, mathematical modelling has sought to distil the essence of complexity into concepts readily understandable by today’s science. One of such approaches has been reported by means of the study of pathways of interaction of biological networks. By concentrating on similar features that biological networks share, it has been recently discovered that at a cellular level, regulation and transcription Networks display certain patterns of connectivity at a much higher rate than expected in an equivalent randomized network. These recurring patterns of interaction, or network “Motifs”, can help us define bread classes of networks and their types of functional elements. In the same way, they can reveal the evolutionary aim by which they have been developed. Network Motifs can be interpreted as structures that have emerged as direct a reflection of the constraints under which the network has evolved. These network Motifs have been found in the biological networks of many systems, suggesting that they are the building blocks of transcription networks [4]. It has been suggested that in biological networks, these recurrent Network Motifs are responsible for carrying out key information processing tasks in the organism [5].

 

From Coupling oscillations and switches in genetic networks.

Switches (bistability) and oscillations (limit cycle) are omnipresent in biological networks. Synthetic genetic networks producing bistability and oscillations have been designed and constructed experimentally. However, in real biological systems, regulatory circuits are usually interconnected and the dynamics of those complex networks is often richer than the dynamics of simple modules. Here we couple the genetic Toggle switch and the Repressilator, two prototypic systems exhibiting bistability and oscillations, respectively. We study two types of coupling. In the first type, the bistable switch is under the control of the oscillator. Numerical simulation of this system allows us to determine the conditions under which a periodic switch between the two stable steady states of the Toggle switch occurs. In addition we show how birhythmicity characterized by the coexistence of two stable small-amplitude limit cycles, can easily be obtained in the system. In the second type of coupling, the oscillator is placed under the control of the Toggleswitch. Numerical simulation of this system shows that this construction could for example be exploited to generate a permanent transition from a stable steady state to self-sustained oscillations (and vice versa) after a transient external perturbation. Those results thus describe qualitative dynamical behaviors that can be generated through the coupling of two simple network modules. These results differ from the dynamical properties resulting from interlocked feedback loops systems in which a given variable is involved at the same time in both positive and negative feedbacks. Finally the models described here may be of interest in synthetic biology, as they give hints on how the coupling should be designed to get the required properties.

 

From Robust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback Loops

To test the generality of the idea that positive feedback enables an oscillator to have a tunable frequency and constant amplitude, we examined several other oscillator models, including five negative feedback–only models: (i) the Goodwin oscillator, a well-studied model relevant to circadian oscillations (18, 19); (ii) the Repressilator, a transcriptional triple-negative feedback loop constructed in Escherichia coli (20); (iii) the “Pentilator,” a Repressilator with five (rather than three) repressors; (iv) the Metabolator (21), a synthetic metabolic oscillator; and (v) the Frzilator, amodel of the control of gliding motions in myxobacteria (22). In four of the cases (Goodwin, Repressilator, Pentilator, and Metabolator), the amplitude/frequency curves were inverted U-shaped curves similar to that seen for the negative feedback–only cell cycle model (Figs. 1B and 3A). In the case of the Frzilator, the legs of the curve were truncated; the oscillator had a nonzero minimal amplitude (Fig. 3A). For all five of the negative feedback–only models, the oscillators functioned over only a narrow range of frequencies (Fig. 3A).

We also examined four positive-plus-negative feedback oscillators: (i) the van der Pol oscillator, inspired by studies of vacuum tubes (12); (ii) the Fitzhugh-Nagumo model of propagating action potentials (23, 24); (iii) the Meyer-Stryer model of calcium oscillations (25); and (iv) a model of circadian oscillations in the cyanobacterial KaiA/B/C system (26–28). In each case, we obtained a flat, wide amplitude/frequency curve (Fig. 3B). Thus, a tunable frequency plus constant amplitude can be obtained from many different positive-plusnegative feedback models; this feature is not peculiar to one particular topology or parameterization.

These findings rationalize why the positiveplus- negative feedback design might have been selected through evolution in cases where a tunable frequency and constant amplitude are important, such as heartbeats and cell cycles. However, it is not clear that an adjustable frequency would be advantageous for circadian oscillations, because frequency is fixed at one cycle per day. Nevertheless, the cyanobacterial circadian oscillator appears to rely on positive feedback (26), and positive feedback loops have been postulated for other circadian oscillators as well (Table 1). This raises the question of whether the positiveplus- negative feedback design might offer additional advantages.

One possibility is that the positive-plusnegative feedback design permits oscillations over a wider range of enzyme concentrations and kinetic constant values, making the oscillator easier to evolve and more robust to variations in its imperfect components. We tested this idea through a Monte Carlo approach.We formulated three simple oscillatormodels: (i) a three-variable triple negative feedback loop with no additional feedback (Fig. 4A), (ii) one with added positive feedback (Fig. 4B), or (iii) one with added negative feedback (Fig. 4C). We generated random parameter sets for the models and then for each set determined whether the model produced limit cycle oscillations.We continued generating parameter sets until we had amassed 500 that gave oscillations.

 

From Robust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback Loops

Sysbio

 

 

Key Terms:

  • Ultra-sensitivity
  • Bi-stability
  • Positive Feedback Loop
  • Negative Feedback Loop
  • Biological Oscillators
  • Biological Switches
  • Biological Networks
  • Network Motifs
  • Regulation Networks
  • Signalling Networks
  • Communication Networks
  • Biological Clocks
  • Circadian Rhythms
  • Harmonic Oscillators
  • Van der Pol Oscillator (Limit Cycle)
  • FitzHugh–Nagumo oscillators (Neural)
  • Limit Cycle Oscillator
  • Cell Cycle
  • Systems Biology
  • Synthetic Biology
  • Gene Regulatory Networks
  • Kuramoto Oscillators
  • Phase Coupled Oscillators
  • Cardic Pacemaker
  • Biochemical Networks
  • Synchronization
  • Goodwin Oscillator
  • Repressilators
  • Fussenegger Oscillators
  • Smolen Oscillators
  • Variable Link Oscillators
  • Metabolators
  • Amplified Negative Feedback Oscillators

 

 

 

Key Sources of Research:

 

 

Ultrasensitivity Part I: Michaelian responses and zero-order ultrasensitivity

James E. Ferrell Jr. and Sang Hoon Ha

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4214216/pdf/nihms-629459.pdf

 

 

 

 

Ultrasensitivity Part II: Multisite phosphorylation, stoichiometric inhibitors, and positive feedback

James E. Ferrell Jr. and Sang Hoon Ha

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4435807/pdf/nihms686079.pdf

 

 

 

Ultrasensitivity part III: cascades, bistable switches, and oscillators

James E. Ferrell Jr and Sang Hoon Ha

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4254632/pdf/nihms635216.pdf

 

 

 

Robust Network Topologies for Generating Switch-Like Cellular Responses

Najaf A. Shah1, Casim A. Sarkar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3121696/pdf/pcbi.1002085.pdf

 

 

 

 

Feedback Loops Shape Cellular Signals in Space and Time

Onn Brandman1 and Tobias Meyer

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2680159/pdf/nihms101299.pdf

 

 

 

Interlinked Fast and Slow Positive Feedback Loops Drive Reliable Cell Decisions

Onn Brandman, James E. Ferrell Jr, Rong Li2,3,4, and Tobias Meyer

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3175767/pdf/nihms180881.pdf

 

 

 

Positive feedback in cellular control systems

Alexander Y. Mitrophanov and Eduardo A. Groisman

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2486260/pdf/nihms-58057.pdf

 

 

 

Effect of positive feedback loops on the robustness of oscillations in the network of cyclin-dependent kinases driving the mammalian cell cycle

Claude Gerard, Didier Gonze and Albert Goldbeter

 

http://onlinelibrary.wiley.com/store/10.1111/j.1742-4658.2012.08585.x/asset/j.1742-4658.2012.08585.x.pdf?v=1&t=j0i1rfq0&s=54814f48d70da4b93bd1632677765a1a5673c8d6

 

 

Design Principles of Biochemical Oscillators

Béla Novak and John J. Tyson

 

 

 

Design principles underlying circadian clocks

D. A. Rand1,†, B. V. Shulgin1, D. Salazar1,2 and A. J. Millar

 

 

 

Positive Feedback Promotes Oscillations in Negative Feedback Loops

Bharath Ananthasubramaniam*, Hanspeter Herzel

 

 

 

Efficient Switches in Biology and Computer Science

Luca Cardelli1,2, Rosa D. Hernansaiz-Ballesteros3, Neil Dalchau1, Attila Csika ́sz-Nagy

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5215766/pdf/pcbi.1005100.pdf

 

 

 

Robust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback Loops

Tony Yu-Chen Tsai,1* Yoon Sup Choi,1,2* Wenzhe Ma,3,4 Joseph R. Pomerening,5 Chao Tang,3,4 James E. Ferrell Jr

https://www.researchgate.net/publication/5253202_Robust_Tunable_Biological_Oscillations_from_Interlinked_Positive_and_Negative_Feedback_Loops?el=1_x_8&enrichId=rgreq-3a45d550364998e0f57384dda12a695f-XXX&enrichSource=Y292ZXJQYWdlOzI0MTY5NjI3MjtBUzoxMzEyODEwMTg5NTM3MjhAMTQwODMxMTI0MjY2OQ==

 

 

 

Biological switches and clocks

John J. Tyson1,*, Reka Albert2, Albert Goldbeter3, Peter Ruoff4 and Jill Sibl

 

http://www.ulb.ac.be/sciences/utc/ARTICLES/2008_Tyson_J_R_Soc_Interface.pdf

https://www.kitp.ucsb.edu/activities/bioclocks07

http://online.kitp.ucsb.edu/online/bioclocks07/

 

 

 

Network thinking in ecology and evolution

Stephen R. Proulx1, Daniel E.L. Promislow2 and Patrick C. Phillips

 

https://pdfs.semanticscholar.org/5665/65601ed2a5c67143b6d4be7193c02235a279.pdf

 

 

 

Networks in ecology

Jordi Bascompte

 

http://izt.ciens.ucv.ve/ecologia/Archivos/ECO_POB%202007/ECOPO7_2007/Bascompte%202007.pdf

 

 

 

Network structure and the biology of populations

Robert M. May

 

https://www.sccs.swarthmore.edu/users/08/bblonder/phys120/docs/may.pdf

 

 

 

Biological networks: Motifs and modules

 

http://bioinfo.vanderbilt.edu/zhanglab/lectures/BMIF310_network_B_Motifs_2009.pdf

 

 

 

Analysis of Biological Networks: Network Motifs

 

http://www.cs.tau.ac.il/~roded/courses/bnet-a06/lec04.pdf

 

 

 

Regulatory networks & Functional motifs

Didier Gonze

 

http://homepages.ulb.ac.be/~dgonze/TEACHING/network_motifs.pdf

 

 

 

Structure and function of the feed-forward loop network motif

S. Mangan and U. Alon

 

http://www.pnas.org/content/100/21/11980.full.pdf

 

 

 

Network Motifs: Simple Building Blocks of Complex Networks

R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, U. Alon

 

http://wilson.med.harvard.edu/nb204/MiloAlon2002.pdf

 

 

 

The dynamics and robustness of Network Motifs in transcription networks

Arturo Araujo

http://www.ucl.ac.uk/~ucbpaar/flies_archivos/Network_Motifs.pdf

 

 

 

Formation of Regulatory Patterns During Signal Propagation in a Mammalian Cellular Network

Avi Ma’ayan, Sherry L. Jenkins, Susana Neves, Anthony Hasseldine, Elizabeth Grace, Benjamin Dubin-Thaler, Narat J. Eungdamrong, Gehzi Weng, Prahlad T. Ram, J. Jeremy Rice, Aaron Kershenbaum, Gustavo A. Stolovitzky, Robert D. Blitzer, and Ravi Iyengar

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3032439/pdf/nihms266526.pdf

 

 

 

Toward Predictive Models of Mammalian Cells

Avi Ma’ayan, Robert D. Blitzer, and Ravi Iyengar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3035045/pdf/nihms266522.pdf

 

 

 

Modeling Cell Signaling Networks

Narat J. Eungdamrong and Ravi Iyengar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3620715/pdf/nihms453834.pdf

 

 

 

Bistability in Biochemical Signaling Models

Eric A. Sobie

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4118931/pdf/nihms-332970.pdf

 

 

An Introduction to Dynamical Systems

Eric A. Sobie

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4118930/pdf/nihms-332968.pdf

 

 

 

Computational approaches for modeling regulatory cellular networks

Narat J. Eungdamrong and Ravi Iyengar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3619405/pdf/nihms-453838.pdf

 

 

Systems Biology—Biomedical Modeling

Eric A. Sobie,* Young-Seon Lee, Sherry L. Jenkins, and Ravi Iyengar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3188945/

 

 

 

Network analyses in systems pharmacology

 

Seth I. Berger and Ravi Iyengar

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2752618/pdf/btp465.pdf

 

 

Biological Networks: The Tinkerer as an Engineer

U Alon

 

http://dp.univr.it/~laudanna/Systems%20Biology/Publications/Reviews/Network%20analysis/Biological%20Networks%20The%20Tinkerer%20as%20an%20Engineer.pdf

 

 

Cell Biology: Networks, Regulation and Pathways

GAŠPER TKACˇ IK , WILLIAM BIALEK

 

https://www.princeton.edu/~wbialek/our_papers/tkacik+bialek_09b.pdf

 

 

 

Coupling oscillations and switches in genetic networks

Didier Gonze

 

https://pdfs.semanticscholar.org/0878/d29052b34bc3fe43649c826fd9fd0506e445.pdf

 

 

 

Biological Oscillators and Switches

 

http://faculty.washington.edu/hqian/amath4-523/Murray-Math-Biol-ch7.pdf

 

 

 

Design principles of biological oscillators

 

Didier Gonze

 Nonlinear Chemical Dynamics: Oscillations, Patterns, and Chaos

 

Irving R. Epstein

Kenneth Showalter

 

 

Modelling biological oscillations

 

Shan He

 

A comparative analysis of synthetic genetic oscillators

 

Oliver Purcell1,*, Nigel J. Savery3, Claire S. Grierson4 and Mario di Bernardo2,5

 

Hierarchy Theory in Biology, Ecology and Evolution

Hierarchy Theory in Biology, Ecology and Evolution

 

I have always been intrigued by multi-level thinking whether it is in organizations, biology, ecology, and evolutionary theory.

  • Plant – Division – Corporate – Industry – Macro-economy
  • Molecules – Organelles – Cells – Tissue – Organs – Whole body
  • Organism – Populations – Communities – Ecosystem –  Bio-Sphere

 

How does human body forms from Molecules?  Is it all evolutionary?  or is there a role for Vitalism?

How to integrate decision making in organizations at multi levels?  From Corporate level to Plant Level.

How does an Individual fits in Groups, Communities, Society, and Ecosystem?

What is the role of fractals thinking in Evolutionary Biology?

 

A SUMMARY OF THE PRINCIPLES OF HIERARCHY THEORY

The Hierarchy theory is a dialect of general systems theory. It has emerged as part of a movement toward a general science of complexity. Rooted in the work of economist, Herbert Simon, chemist, Ilya Prigogine, and psychologist, Jean Piaget, hierarchy theory focuses upon levels of organization and issues of scale. There is significant emphasis upon the observer in the system.

Hierarchies occur in social systems, biological structures, and in the biological taxonomies. Since scholars and laypersons use hierarchy and hierarchical concepts commonly, it would seem reasonable to have a theory of hierarchies. Hierarchy theory uses a relatively small set of principles to keep track of the complex structure and a behavior of systems with multiple levels. A set of definitions and principles follows immediately:

Hierarchy: in mathematical terms, it is a partially ordered set. In less austere terms, a hierarchy is a collection of parts with ordered asymmetric relationships inside a whole. That is to say, upper levels are above lower levels, and the relationship upwards is asymmetric with the relationships downwards.

Hierarchical levels: levels are populated by entities whose properties characterize the level in question. A given entity may belong to any number of levels, depending on the criteria used to link levels above and below. For example, an individual human being may be a member of the level i) human, ii) primate, iii) organism or iv) host of a parasite, depending on the relationship of the level in question to those above and below.

Level of organization: this type of level fits into its hierarchy by virtue of set of definitions that lock the level in question to those above and below. For example, a biological population level is an aggregate of entities from the organism level of organization, but it is only so by definition. There is no particular scale involved in the population level of organization, in that some organisms are larger than some populations, as in the case of skin parasites.

Level of observation: this type of level fits into its hierarchy by virtue of relative scaling considerations. For example, the host of a skin parasite represents the context for the population of parasites; it is a landscape, even though the host may be seen as belonging to a level of organization, organism, that is lower than the collection of parasites, a population.

The criterion for observation: when a system is observed, there are two separate considerations. One is the spatiotemporal scale at which the observations are made. The other is the criterion for observation, which defines the system in the foreground away from all the rest in the background. The criterion for observation uses the types of parts and their relationships to each other to characterize the system in the foreground. If criteria for observation are linked together in an asymmetric fashion, then the criteria lead to levels of organization. Otherwise, criteria for observation merely generate isolated classes.

The ordering of levels: there are several criteria whereby other levels reside above lower levels. These criteria often run in parallel, but sometimes only one or a few of them apply. Upper levels are above lower levels by virtue of: 1) being the context of, 2) offering constraint to, 3) behaving more slowly at a lower frequency than, 4) being populated by entities with greater integrity and higher bond strength than, and 5), containing and being made of – lower levels.

Nested and non-nested hierarchies: nested hierarchies involve levels which consist of, and contain, lower levels. Non-nested hierarchies are more general in that the requirement of containment of lower levels is relaxed. For example, an army consists of a collection of soldiers and is made up of them. Thus an army is a nested hierarchy. On the other hand, the general at the top of a military command does not consist of his soldiers and so the military command is a non-nested hierarchy with regard to the soldiers in the army. Pecking orders and a food chains are also non-nested hierarchies.

Duality in hierarchies: the dualism in hierarchies appears to come from a set of complementarities that line up with: observer-observed, process-structure, rate-dependent versus rate-independent, and part-whole. Arthur Koestler in his “Ghost in The Machine” referred to the notion of holon, which means an entity in a hierarchy that is at once a whole and at the same time a part. Thus a holon at once operates as a quasi-autonomous whole that integrates its parts, while working to integrate itself into an upper level purpose or role. The lower level answers the question “How?” and the upper level answers the question, “So what?”

Constraint versus possibilities: when one looks at a system there are two separate reasons behind what one sees. First, it is not possible to see something if the parts of the system cannot do what is required of them to achieve the arrangement in the whole. These are the limits of physical possibility. The limits of possibility come from lower levels in the hierarchy. The second entirely separate reason for what one sees is to do with what is allowed by the upper level constraints. An example here would be that mammals have five digits. There is no physical reason for mammals having five digits on their hands and feet, because it comes not from physical limits, but from the constraints of having a mammal heritage. Any number of the digits is possible within the physical limits, but in mammals only five digits are allowed by the biological constraints. Constraints come from above, while the limits as to what is possible come from below. The concept of hierarchy becomes confused unless one makes the distinction between limits from below and limits from above. The distinction between mechanisms below and purposes above turn on the issue of constraint versus possibility. Forget the distinction, and biology becomes pointlessly confused, impossibly complicated chemistry, while chemistry becomes unwieldy physics.

Complexity and self-simplification: Howard Pattee has identified that as a system becomes more elaborately hierarchical its behavior becomes simple. The reason is that, with the emergence of intermediate levels, the lowest level entities become constrained to be far from equilibrium. As a result, the lowest level entities lose degrees of freedom and are held against the upper level constraint to give constant behavior. Deep hierarchical structure indicates elaborate organization, and deep hierarchies are often considered as complex systems by virtue of hierarchical depth.

Complexity versus complicatedness: a hierarchical structure with a large number of lowest level entities, but with simple organization, offers a low flat hierarchy that is complicated rather than complex. The behavior of structurally complicated systems is behaviorally elaborate and so complicated, whereas the behavior of deep hierarchically complex systems is simple.

Hierarchy theory is as much as anything a theory of observation. It has been significantly operationalized in ecology, but has been applied relatively infrequently outside that science. There is a negative reaction to hierarchy theory in the social sciences, by virtue of implications of rigid autocratic systems or authority. When applied in a more general fashion, even liberal and non-authoritarian systems can be described effectively in hierarchical terms. There is a politically correct set of labels that avoid the word hierarchy, but they unnecessarily introduce jargon into a field that has enough special vocabulary as it is.

A SHORT ANNOTATED BIBLIOGRAPHY OF HIERARCHY THEORY.

This bibliography is in chronological order, so that the reader can identify the early classics as opposed to the later refinements. If you must choose just one book to read, turn to the last reference in this bibliography, Ahl and Allen, 1996. Simon, H.. A. 1962. The architecture of complexity. Proceedings of the American philosophical society 106: 467-82. This is the foundation paper of hierarchy theory originating from an economist. It was a re-published in “Sciences of the Artificial” by Simon. It introduces the idea of near-decomposability. If systems were completely decomposable, then there would be no emergent whole, because the parts would exist only separately. The “near” in near-decomposable allows the upper level to emerge from the fact that the parts anre not completely separate.

Koestler, Arthur. 1967. The ghost in the machine. Macmillan, New York. This is a long hard look at human social structure in hierarchical terms. The notion of holon first occurs in this work. This is a classic work, but is easily accessible to the lay public.

Whyte, L.. L.., A. G. Wilson and D. Wilson (eds.). 1969. Hierarchical structures. American Elsevier, New York. This is a classic collection of early scholarly works by some of the founders of hierarchical thinking.

Pattee, H.. H. (ed.) 1973. Hierarchy theory: the challenge or complex systems. Braziller, New York. This edited volume has some classic articles by Pattee, Simon and others.

Allen, T. F. H. and T. B. Starr. 1982. Hierarchy: perspectives for ecological complexity. University Chicago Press. This book has a significant ecological component but is much more generally about hierarchical structure. It is abstract and a somewhat technical treatment but has been the foundation work for the application of hierarchy theory in ecology and complex systems theory at large.

Salthe, S. 1985. Evolving Hierarchical Systems: their structure and representation. Columbia University Press, New York. This book has a strong structural bias, in contrast to the process oriented approach of Allen and the other ecologists in this bibliography. Salthe introduces the notion of the Triadic, where there is a focus on 1) the system as both a whole above the levels below and 2) a part belonging to another level above, 3) not forgetting the level of the structure itself in between. While much biological hierarchy theory takes an anti-realist point view, or is at least reality-agnostic, wherein the ultimate reality of hierarchical arrangement is left moot, Salthe’s version of hierarchy theory is concerned with the ultimate reality of structure. The anti-realist view of structure is that it is imposed by the observer, and may or may not correspond to any ultimate reality. If structure does correspond to ultimate, external reality, we could never know that to be so. Salthe’s logic is consistent but always takes a structural and ontological position.

O’Neill, R. V., D. DeAngelis, J. Waide and T. F. H. Allen. 1986. A hierarchical concept of ecosystems. Princeton University Press. This is a distinctly ecological application of hierarchy theory, making the critical distinction between process functional ecosystem approaches as opposed to population and community relationships. It is an application of hierarchy theory to ecosystem analysis.

Allen T. F. H. and T. Hoekstra. 1992. Toward a unified ecology. Columbia University Press. This book turns on hierarchy theory, but is principally a book about ecology. It goes beyond the O’Neill et al book, in that it makes the distinction between many types of ecology (landscape, ecosystem, community, organism, population, and biomes) on the one hand, and scale of ecology on the other hand. It ends with practical applications of hierarchy theory and ecological management.

Ahl, V. and T. F. H. Allen. 1996. Hierarchy theory, a vision, vocabulary and epistemology. Columbia University Press. This slim a volume is an interdisciplinary account of a hierarchy theory, and represents the shallow end of the pool. It is the primer version of Allen and Starr 1982. It is full of graphical images to ease the reader into a hierarchical perspective. It makes the distinction between levels of organization and levels of observation. It takes a moderate anti-realist point of view, wherein there may be an external reality, but it is not relevant to the discourse. We only have access to experience, which must of necessity involve observer values and subjectivity. There are examples from a wide discussion of many disciplines. Included are examples from psychology, ecology, the law, political systems and philosophy. It makes reference to the global and technological problems facing humanity, and offers hierarchy theory as one tool in the struggle. The summary of hierarchy theory in the opening paragraphs above comes from this book.

This summary was compiled by

Timothy F. Allen, Professor of Botany,
University of Wisconsin Madison,
Madison Wisconsin 53706 — 1381.
Email – tfallen@facstaff.wisc.edu

 

 

Key People:

  • James Grier Miller
  • Howard Pattee
  • Stanley Salthe
  • T F Allen
  • Herbert Simon
  • NILES ELDREDGE
  • CS Holling

 

 

Key Sources of Research:

 

A SUMMARY OF THE PRINCIPLES OF HIERARCHY THEORY

T Allen

http://www.isss.org/hierarchy.htm

http://www.botany.wisc.edu/allenlab/AllenLab/Hierarchy.html

 

 

Hierarchy Theory

Paweł Leśniewski

 

http://www.uni-kiel.de/ecology/users/fmueller/salzau2006/ea_presentations/Data/2006-06-28_-_Hierarchy_Theory.pdf

 

 

Summary of the Principles of Hierarchy Theory

S.N. Salthe

 

http://www.nbi.dk/~natphil/salthe/Summary_of_the_Principles_o.pdf

 

 

HOWARD PATTEE’S THEORETICAL BIOLOGY:

A RADICAL EPISTEMOLOGICAL STANCE TO APPROACH LIFE, EVOLUTION ANDCOMPLEXITY.

Jon Umerez

 

http://www.informatics.indiana.edu/rocha/publications/pattee/umerez.pdf

 

 

 

Hierarchy Theory as the Formal Basis of Evolutionary Theory

 

http://www.bbk.ac.uk/tpru/StephenWood/Publications/HierarchyTheoryastheFormalBasisofEvolutionaryTheory.pdf

 

 

The Concept of Levels of Organization in the Biological Sciences

 

PhD Thesis Submitted August 2014 Revised June 2015

Daniel Stephen Brooks

 

http://d-nb.info/1082033960/34

 

 

A spatially explicit hierarchical approach to modeling complex ecological systems: theory and applications

Jianguo Wu , John L. David

 

http://leml.asu.edu/jingle/Web_Pages/Wu_Pubs/PDF_Files/Wu_David_2002.PDF

 

 

What is the Hierarchy Theory of Evolution?

 

http://hierarchygroup.com/wp-content/uploads/2014/07/What-Is-The-Hierarchy-Theory.pdf

 

 

HIERARCHICAL ORGANIZATION OF ECOSYSTEMS

Jackson R. Webster

 

http://coweeta.uga.edu/publications/274.pdf

 

 

Ecological hierarchies and self-organisation – Pattern analysis, modelling and process integration across scales

Hauke Reutera,, Fred Jopp, José M. Blanco-Morenod, Christian Damgaarde, Yiannis Matsinosf, Donald L. DeAngelis

 

http://izt.ciens.ucv.ve/ecologia/Archivos/ECO_POB%202010/ECOPO1_2010/Reuter_etal_BAAE%202010.pdf

 

 

Levels of organization in biology: on the nature and nomenclature of ecology’s fourth level

William Z. Lidicker, Jr

 

http://www.uff.br/ecosed/Artigo4.pdf

 

 

Chapter 24

Hierarchy Theory: An Overview

Jianguo Wu

 

http://izt.ciens.ucv.ve/ecologia/Archivos/ECO_POB%202016/ECOPO7_2016/Jorgensen%20et%20al%202016.pdf

 

 

Heterarchies: Reconciling Networks and Hierarchies

Graeme S. Cumming

https://www.researchgate.net/publication/303508940_Heterarchies_Reconciling_Networks_and_Hierarchies

 

 

Evolutionary Theory

A HIERARCHICAL PERSPECTIVE

EDITED BY NILES ELDREDGE, TELMO PIEVANI, EMANUELE SERRELLI, AND ILYA TEMKIN

 

 

Holons, creaons, genons, environs, in hierarchy theory: Where we have gone

Timothy Allen, Mario Giampietro

http://www.sciencedirect.com/science/article/pii/S0304380014002993

 

 

The Evolutionary Foundations of Hierarchy: Status, Dominance, Prestige, and Leadership

Mark van Vugt & Joshua M. Tybur

http://www.professormarkvanvugt.com/images/files/Handbook_of_Evolutionary_Psychologymvv2014rev.pdf

 

 

The Microfoundations of Macroeconomics: An Evolutionary Perspective

Jeroen C.J.M. van den Bergh

John M. Gowdy

 

https://papers.tinbergen.nl/00021.pdf

 

 

Understanding the complexity of Economic, Ecological, and Social Systems

C S Holling

http://www.esf.edu/cue/documents/Holling_Complexity-EconEcol-SocialSys_2001.pdf

 

 

Hierarchical Structures

Stanley N. Salthe

 

https://www.researchgate.net/profile/Salthe_Stanley/publication/257522907_Hierarchical_Structures/links/5768411408ae7f0756a2248c.pdf

 

 

Two Frameworks for Complexity Generation in Biological Systems

Stanley N. Salthe

 

http://www.nbi.dk/natphil/salthe/A-life_Conf_paper_Word.pdf

http://www.nbi.dk/~natphil/salthe/_publ_classified_by_topic.pdf

 

 

Spatial scaling in ecology

J. A. WIENS

 

http://www.functionalecology.org/SpringboardWebApp/userfiles/fec/file/Spatial%20scaling%20in%20ecology%20v3%20n4.pdf

 

 

The Spirit of Evolution

by Roger Walsh

An overview of Ken Wilber’s book Sex, Ecology, Spirituality: The Spirit of Evolution (Shambhala, 1995).

http://cogweb.ucla.edu/CogSci/Walsh_on_Wilber_95.html

Shapes and Patterns in Nature

Shapes and Patterns in Nature

 

There are so many colors, shapes, and patterns in nature.

  • Seashells
  • Animal Skins (Zebra, Leopard)
  • Butterflies
  • Shape of Plants
  • Flowers (Sun Flower)
  • Fruits (Pineapple)

How do we explain these from perspective of science?  There are several branches of science which have explored these questions for decades.  There are Reaction Diffusion Models and Cellular Automata models explaining development of patterns on seashells, plants and animal skins.  There is L-system developed by Aristid Lindenmayer to explain development of plants.  It is a fascinating subject.

 

From Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

There are several studies have been carried out in a number of scientific disciplines, such as mathematics, biology, paleontology and computer engineering to understand and decipher the relations of the seashells complex forms. Starting with Descartes, Figure 4 shows a time line in which many investigators having focused on the curves of these shells and their mathematical properties. They all outlined a number of mathematical relations that control the overall geometry of seashells.

After examining the existing seashell models in literature it is seen that they all followed Raup’s model which roughly abstracts the seashell form using three parameters; whorl (rate of expansion of the generating curve), distance (relative distance between the generating curve and axis of coiling), and translation (the change of the cone’s movement along an axis with respect to the whorl), an ellipse as the whorl cross-section as well. However, it is clear from the observations of actual shells (Figure 5) that the cross-section is more complex than the input that the three parameters allow. In the pursuit of realistic visualizations, Kawaguchi enhanced the appearance of shell models using filled polygons which represented the surface of shells more convincingly than line drawings. Similar techniques were used subsequently by Oppenheimer (1986). A different approach was adopted by Pickover (1989) who approximated shell surfaces by using interpenetrating spheres. Illert (1989) introduced Frenet Frames (Bronsvoort, 1985) to precisely orient the opening of a shell. His model also captured a form of surface sculpture. Cortie (1989) studied the pattern forms on the surface of the shell model (Meinhardt, 2003). Finally, the model of seashell geometry by Fowler et al. (2003) was similar to that introduced by Raup, and was the first to implement free-form cross sections using a Bézier curve (Farin, 2002 Rogers, 2001) as the input. It can be claimed that, studies above all focused on modeling the appearance of the shell surface.

All these approaches can be considered as a milestone for their era, as each model reflects the observation and tools of measurement, modeling and technologies of their time. In all these approaches seashells were modeled as a single surface, as a twodimensional object, and embedded in three-dimensional space. Today, such modeling research should be carried out employing observation tools, knowledge, information, and computational technologies to the maximum extent. For this reason, we developed a mathematical model that can be transformed into a computational model for further studies (such as overall behavior of shells, form-structure relations, form finding explorations etc.) to explore potentials of such optimized forms.

 

From Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

 

patterns

 

From Computational models of plant development and form

A broad program of using mathematical reasoning in the study of the development and form of living organisms was initiated almost 100 yr ago by D’Arcy Thompson (1942) in his landmark book On Growth and Form (see Keller, 2002, for a historical analysis). One of his most influential contributions was the ‘theory of transformations’, which showed how forms of different species could be geometrically related to each other. The theory of transformations was extended to relate younger and older forms of a developing organism (Richards & Kavanagh, 1945), but did not incorporate the formation and differentiation of new organs. This limitation was addressed a quarter of a century later by Lindenmayer (1968, 1971), who introduced an original mathematical formalism, subsequently called L-systems, to describe the development of linear and branching structures at the cellular level. By the mid 1970s, computational models based on Lsystems and other formalisms had been applied to study several aspects of plant development, including the development of leaves and inflorescences, and the formation of phyllotactic patterns (Lindenmayer, 1978). The questions being asked included the impact of distinct modes of information transfer (lineage vs interaction) on plant development, and the relationship between local development and global form. Similar interests underlied the independent pioneering work of Honda and co-workers on the modeling of trees (Honda, 1971; Borchert & Honda, 1984).

Another class of models was pioneered by Turing (1952), who showed mathematically that, in a system of two or more diffusing reagents, a pattern of high and low concentrations may spontaneously emerge from an initially uniform distribution. This was a surprising result, as it appeared to contradict the second law of thermodynamics: the general tendency of systems to proceed from more organized states toward disorder (the apparent paradox is resolved by jointly considering the reaction–diffusion system and its surroundings). Related models were introduced, under the name of activator–inhibitor and activator-substrate (depletion) systems, by Gierer & Meinhardt (1972), and extensively investigated by Meinhardt (1982). Reaction–diffusion systems showed how, in principle, molecular-level interactions may lead to morphogenesis and differentiation. In plants, reaction– diffusion-type models have been used to explain the patterning of trichomes in leaves and hair cells in roots (Digiuni et al., 2008; Savage et al., 2008; Jo¨nsson & Krupinski, 2010; Benı´tez et al., 2011). Nevertheless, the extent to which reaction–diffusion models apply to the plant kingdom appears to be limited (Kepinski & Leyser, 2005; Berleth et al., 2007). A significant role is played instead by mechanisms involving active transport of the plant hormone auxin (Section V). In some cases, such as the generation of phyllotactic patterns, this reliance on active transport is difficult to explain in evolutionary terms, as reaction–diffusion systems can generate the same patterns. Spatio-temporal coordination of other developmental processes, however, such as bud activation, requires long-distance signaling. Active transport may thus have evolved to overcome the limitations of diffusion, which is very slow over long distances (Crick, 1971).

In the last decade, computational modeling has become a mainstream technique in developmental plant biology, as reflected in numerous reviews (e.g. Prusinkiewicz, 2004b; Prusinkiewicz & Rolland-Lagan, 2006; Grieneisen & Scheres, 2009; Chickarmane et al., 2010; Jo¨nsson&Krupinski, 2010; Jo¨nsson et al., 2012). On the one hand, the sequencing of the human genome put in focus the chasm between knowing the genome of an organism and understanding how this organismdevelops and functions.Computational models bridge this chasm. On the other hand, successes of early conceptual models that relate patterns of gene expression to the form of animals (Lawrence, 1992) and plants (Coen & Meyerowitz, 1991) have prompted a quest for a comprehensive, mechanistic understanding of development (Coen, 1999). Current experimental techniques for tracking growth and observing marked proteins in living tissues (Reddy et al., 2004; Fernandez et al., 2010) are yielding a wealth of data that correlate molecular-level processes with plant development and form. Computational models play an increasingly important role in interpreting these data.

The use of models has been accelerated by the advancements in computer hardware, software, and modeling methodologies. General-purpose mathematical software (e.g. Mathematica and MATLAB), modeling programs built on the basis of this software (e.g. GFtbox, Kennaway et al., 2011) and specialized packages for modeling plants (e.g. the Virtual Laboratory and L-studio (Prusinkiewicz, 2004a), OpenAlea (Pradal et al., 2008) and VirtualLeaf (Merks et al., 2011)) facilitate model construction, compared with general-purpose programming languages. Furthermore, current computers are sufficiently fast to simulate and visualize many models at interactive or close-to-interactive rates, which is convenient for model exploration.

 

From The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin

In his paper entitled ‘The chemical basis of morphogenesis’ Turing presented a ground-breaking idea that a combination of reaction and diffusion can generate spatial patterns (Turing 1952). In the paper, he studied the behaviour of a complex system in which two substances interact with each other and diffuse at different diffusion rates, which is known as the reaction–diffusion (RD) system. Turing proved mathematically that such system is able to form some characteristic spatio-temporal patterns in the field. One of the most significant deviations is s formation of a stable periodic pattern. He stated that the spatial pattern generated by the system might provide positional information for a developing embryo.

In spite of the importance of the idea in the developmental biology, his model was not accepted by most experimental biologists mainly because there were no experimental technologies available to test it. Therefore, most of those who took over and developed the Turing’s idea were applied mathematicians and physicists. They proposed various types of model that developed Turing’s original equation to fit real, naturally occurring phenomena (Meinhardt 1982; Murray & Myerscough 1991; Murray 1993; Nagorcka & Mooney 1992). Although the equations for each model differ, they all share the basic requirement of the original model; that is, ‘waves’ are made from the interactions of two putative chemical substances which we refer to here as the ‘activator’ and the ‘inhibitor’ (Meinhardt 1982).

 

Key Terms

  • Development Biology
  • Mathematical Biology
  • Biomathematics
  • Morphogenesis
  • Phyllotaxis
  • Evolutionary Biology
  • Nonlinear dynamical systems
  • Cellular Automata
  • Fractals
  • Iterated Systems
  • L-Systems
  • Pattern Formation
  • IFS (Iterated Functions Set)
  • Theoretical Biology
  • diffusion–reaction (DR) model
  • Systems Biology
  • Code Biology
  • Computational Biology
  • Algorithmic Biology
  • Complex Systems
  • Turing Patterns

 

 

Key People:

  • D’Arcy Wentworth Thompson
  • Aristid Lindenmayer
  • Alan Turing
  • Hans Meinhardt
  • Philip Ball
  • Przemyslaw Prusinkiewicz
  • Murray JD
  • Stephen Wolfram

 

 

Key Sources of Research:

 

On Growth and Form

Thompson D’Arcy W.

(1952)

 

 

The Algorithmic Beauty of Plants

Prusinkiewicz, Przemyslaw, Lindenmayer, Aristid

 

 

The Algorithmic Beauty of Seashells

Meinhardt H, Prusinkiewicz P, Fowler D

(2003)

(Springer, New York), 3rd Ed.

 

 

The Algorithmic Beauty of Seaweeds, Sponges and Corals

Kaandorp, Jaap A., Kübler, Janet E.

 

 

Mathematical Biology

Murray JD

(2003)

 

 

Models of biological pattern formation

Meinhardt H

(1982)

 

 

The chemical basis of morphogenesis.

Turing A

(1952)

http://cba.mit.edu/events/03.11.ASE/docs/Turing.pdf

 

 

Pattern formation by coupled oscillations: The pigmentation patterns on the shells of molluscs

Hans Meinhardt, Martin Klingler

 

 

The Self-Made Tapestry Pattern formation in nature

Philip Ball

1999

 

 

Models of biological pattern formation in space and time

Hans Meinhardt

2014

http://www.euresisjournal.org/public/article/pdf/Meinhardt.pdf

 

 

Models of biological pattern formation

Hans Meinhardt,

http://www.ulb.ac.be/sciences/leurre/Pise-particles03/Hans_Meinhardt.pdf

 

 

 

Cellular Automata, PDEs, and Pattern Formation

 

https://arxiv.org/pdf/1003.1983.pdf

 

 

The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

By Gary William Flake

 

 

The Curves of Life

Cook, T

1979

Dover Publications, Inc. New York.

 

 

Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation

Shigeru Kondo1* and Takashi Miura

2010

http://scienceintheclassroom.org/sites/default/files/related/reaction-diffusion_model_as_a_framework_for_understanding_biological_pattern_formation.pdf

http://www.lps.ens.fr/~hakim/BIBLIOM2ICFPCOURS/kondomiura10science.pdf

 

 

The Hegemony of Molecular Biology

PHILIP KITCHER

 

http://joelvelasco.net/teaching/2890/kitcher99-hegemony.pdf

 

 

Modeling seashells

 

Deborah R. Fowlery􏰣, Hans Meinhardtz and Przemyslaw Prusinkiewicz

http://algorithmicbotany.org/papers/shells.sig92.pdf

 

 

The neural origins of shell structure and pattern in aquatic mollusks

Alistair Boettigera, Bard Ermentroutb, and George Oster

2009

http://www.pnas.org/content/106/16/6837.full.pdf

 

 

Mechanical basis of morphogenesis and convergent evolution of spiny seashells

Régis Chirata, Derek E. Moultonb,1, and Alain Goriely

2013

 

http://www.pnas.org/content/110/15/6015.full.pdf

 

 

The Geometry and Pigmentation of Seashells

S Coombes

2009

https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf

 

 

PATTERNS IN NATURE

Richie Khandelwal

Sahil Sahni

 

http://www.dstuns.iitm.ac.in/teaching-and-presentations/teaching/undergraduate%20courses/vy305-molecular-architecture-and-evolution-of-functions/presentations/presentations-2006/P7.pdf

 

 

Forging patterns and making waves from biology to geology: a commentary on Turing (1952) ‘The chemical basis of morphogenesis’

Philip Ball

 

https://pdfs.semanticscholar.org/ecec/f989a13264a455ec2898ed361b1c435b5f0c.pdf

 

 

Mollusc Shell Pigmentation: Cellular Automaton Simulations and Evidence for Undecidability

INGO KUSCH AND MARIO MARKUS

1995

 

http://ca4ls.wdfiles.com/local–files/material-2013/KuschMarkus1996.pdf

 

 

Pattern Formation in Reaction-Diffusion Systems

Masayasu Mimura

 

https://pdfs.semanticscholar.org/b5d1/7adbe7e696d4ba9ad3a89fed4ba15549a091.pdf

 

 

The Natural 3D Spiral

Gur Harary and Ayellet Tal

 

http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf

 

 

A Model for Pattern Formation on the Shells of Molluscs

 

 

 

The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin

 

Shigeru Kondo

http://hopf.chem.brandeis.edu/members_content/yanglingfa/pattern/Turing/The%20reaction-diffusion%20system_%20a%20mechanism%20for%20autonomous.pdf

 

 

Mechanical growth and morphogenesis of seashells

Derek E. Moulton, Alain Goriely and R ́egis Chirat

 

http://eprints.maths.ox.ac.uk/1476/1/finalOR01.pdf

 

 

Scaling of morphogenetic patterns in continuous and discrete models

 

http://repository.liv.ac.uk/17293/1/RasolonjanaharyMan_Sep2013_17293.pdf

 

 

On the Dynamics of a Forced Reaction-Diffusion Model for Biological Pattern Formation

A A Tsonis, JB Elsner, P A Tsonis

http://myweb.fsu.edu/jelsner/PDF/Research/TsonisElsnerTsonis1989.pdf

 

 

 

A Model for Pattern Formation on the Shells of Molluscs

H M

http://ikitprod.free.fr/uploads/M1_doc_graph/5_doc.pdf

 

 

Impact of Turing’s Work

Maini

https://people.maths.ox.ac.uk/maini/PKM%20publications/172.pdf

 

 

The possible role of reaction–diffusion in leaf shape

Nigel R. Franks1* and Nicholas F. Britton

 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1690673/pdf/10972123.pdf

 

 

Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism

Motoomi Yamaguchi*†, Eiichi Yoshimoto‡, and Shigeru Kondo

 

http://www.fbs.osaka-u.ac.jp/labs/skondo/paper/yamaguchi2007.pdf

 

 

Turing Patterns

P Ball

http://www.philipball.co.uk/images/stories/docs/pdf/Turing_long.pdf

 

 

 

MODELS FOR PIGMENT PATTERN FORMATION IN THE SKIN OF FISHES

K.J. PAINTER

http://www.macs.hw.ac.uk/~painter/articles/kjp006.pdf

 

 

 

Web Resource for Algorithmic Botony 

http://algorithmicbotany.org/papers/

 

 

FRACTAL GEOMETRY AND SUPERFORMULA TO MODEL NATURAL SHAPES

Nicoletta Sala

2013

http://www.arpapress.com/volumes/vol16issue4/ijrras_16_4_09.pdf

 

 

The Geometry of Seashells

Dr S Coombes

 

http://simetria.dim.uchile.cl/matematico/extras/SeaShells.pdf

 

 

SEASHELLS: THE PLAINNESS AND BEAUTY OF THEIR MATHEMATICAL DESCRIPTION

JORGE PICADO

 

http://www.mat.uc.pt/~picado/conchas/eng/article.pdf

 

 

Models for the morphogenesis of the molluscan shell

 

http://folk.uio.no/oyviham/molluscanshell.pdf

 

 

Modeling Seashell Morphology

 

http://academics.smcvt.edu/jellis-monaghan/papers/papers/teaching%20papers/AE-MKMpre.pdf

 

 

Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

 

http://papers.cumincad.org/data/works/att/ecaade2009_164.content.pdf

 

 

The Neural Origins of Sea Shell Patterns

http://redwood.berkeley.edu/bruno/public/Shells.pdf

 

 

Biological Pattern Formation : from Basic Mechanisms to Complex Structures

A. J. Kochy and H. Meinhardt

 

 

Form-Optimizing in Biological Structures The Morphology of Seashells

EDGAR STACH University of Tennessee

 

 

A Theory of Biological Pattern Formation

A. Gierer and H. Meinhardt

1972

 

http://jeti.uni-freiburg.de/studenten_seminar/term_paper_WS_16_17/gierer_meinhardt.pdf

 

 

Cellular Automata as Models of Complexity

Stephen Wolfram,

Nature 311 (5985): 419–424, 1984

http://gbic.biol.rug.nl/~rbreitling/dagstuhlpublications/006_Wolfram1984.pdf

 

 

Website on Oliva Porphyria

http://oliva.porphyria.free.fr/menu%20GB.html

 

 

Evolution of patterns on Conus shells

Zhenqiang Gonga, Nichilos J. Matzkeb, Bard Ermentroutc, Dawn Songa, Jann E. Vendettib, Montgomery Slatkinb, and George Oster

 

https://people.eecs.berkeley.edu/~dawnsong/papers/2012%20Evolution%20of%20patterns%20on%20Conus%20shells%20_E234.full.pdf

 

 

Theoretical aspects of pattern formation and neuronal development

http://www.eb.tuebingen.mpg.de/de/forschung/emeriti/hans-meinhardt/home.html

 

 

20+ Photos Of Geometrical Plants For Symmetry Lovers

http://www.boredpanda.com/geometry-symmetry-plants-nature/

 

 

Computational models of plant development and form

Przemyslaw Prusinkiewicz and Adam Runions

 

http://www.sns.ias.edu/pitp2/2012files/tansley.np2012.pdf

 

 

Periodic pattern formation in reaction–diffusion systems: An introduction for numerical simulation

Takashi Miura* and Philip K. Maini

 

https://people.maths.ox.ac.uk/maini/PKM%20publications/173.pdf

 

 

Dynamics of Complex Systems

Yaneer Bar-yam

http://necsi.edu/publications/dcs/index.html#fulltext