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




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





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

James E. Ferrell Jr. and Sang Hoon Ha




Ultrasensitivity part III: cascades, bistable switches, and oscillators

James E. Ferrell Jr and Sang Hoon Ha




Robust Network Topologies for Generating Switch-Like Cellular Responses

Najaf A. Shah1, Casim A. Sarkar





Feedback Loops Shape Cellular Signals in Space and Time

Onn Brandman1 and Tobias Meyer




Interlinked Fast and Slow Positive Feedback Loops Drive Reliable Cell Decisions

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




Positive feedback in cellular control systems

Alexander Y. Mitrophanov and Eduardo A. Groisman




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



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




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




Biological switches and clocks

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




Network thinking in ecology and evolution

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




Networks in ecology

Jordi Bascompte




Network structure and the biology of populations

Robert M. May




Biological networks: Motifs and modules




Analysis of Biological Networks: Network Motifs




Regulatory networks & Functional motifs

Didier Gonze




Structure and function of the feed-forward loop network motif

S. Mangan and U. Alon




Network Motifs: Simple Building Blocks of Complex Networks

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




The dynamics and robustness of Network Motifs in transcription networks

Arturo Araujo




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




Toward Predictive Models of Mammalian Cells

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




Modeling Cell Signaling Networks

Narat J. Eungdamrong and Ravi Iyengar




Bistability in Biochemical Signaling Models

Eric A. Sobie



An Introduction to Dynamical Systems

Eric A. Sobie




Computational approaches for modeling regulatory cellular networks

Narat J. Eungdamrong and Ravi Iyengar



Systems Biology—Biomedical Modeling

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




Network analyses in systems pharmacology


Seth I. Berger and Ravi Iyengar



Biological Networks: The Tinkerer as an Engineer

U Alon



Cell Biology: Networks, Regulation and Pathways





Coupling oscillations and switches in genetic networks

Didier Gonze




Biological Oscillators and Switches




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


Growth and Form in Nature: Power Laws and Fractals

Growth and Form in Nature: Power Laws and Fractals


There are several instances of power laws found in nature and in society.  Some of the well known ones are:

  • City Sizes (Zipf’s Law)
  • Firm Sizes
  • Stock Market Movements
  • Income and Wealth (Pareto’s Law)
  • Metabolic Rate and Body Mass (Kleiber’s Law-3/4 or Rubner’s Law-2/3)


Power laws and Scaling in Biology

After 1997 paper by West et all, many publications have analyzed  empirical evidence as to what the correct exponent is and what is the fundamental theoretical basis for power law.

West found 3/4 as exponent, others have reported 1/4, 2/3, 4/5 etc.

Animals and Mammals follow 3/4 exponent.  Plants follow 2/3.

The Metabolic Theory of Ecology

Scaling in biology has a rich and important history. Typically body mass, or some other parameter relating to organism size, is related to anatomical, physiological, and ecological parameters across species. Quite remarkably, diverse organisms, from tiny microbes to the earth’s largest organisms are found to fall along a common slope, with a high degree of variance explained. The beauty of such scaling ‘‘laws’’ has been the generality in biotic organization that they suggest, and the challenge (for ecologists) has often been interpreting their mechanistic bases and ecological consequences.

Scaling laws have thus far inspired scientists in at least three major areas. First, scaling laws may illuminate biology that is otherwise shrouded. For example, if scaling relationships can account for variation in a parameter of interest, the residual variation may be much more easily examined because the major influence of some trait, say, body size, is removed. Second, some scientists have taken an interest in ‘‘the exponent’’—essentially the exponential scaling values that produce the allometric relationship. What are the precise values of these exponents? Are they all from a family of particular values (quarter powers) for many different biological relationships? This area seeks to define the generality of patterns in nature and to explore the empirical robustness of the relationships. Third, from a mechanistic perspective, if scaling laws are mechanistic and truly general, then this suggests some underlying common biological process that forms the structure and function of species and ultimately generates biological diversity. The mechanistics of scaling from metabolism and the currently favored fractal network model of resource acquisition and allocation may allow scientists to understand the laws of how life diversified and is constrained. Perhaps more importantly, such a mechanistic understanding should allow the successful prediction of evolutionary trends, responses of organisms to global change, and other basic and applied biological problems.

The Ecological Society of America’s MacArthur Award winner, James H. Brown, working together with colleagues for over a decade on scaling in biology, has arrived at an outline for a metabolic theory of ecology—a proposal for a unifying theory employing one of the most fundamental aspects of biology, metabolism. This metabolic theory incorporates body size, temperature (metabolic kinetics described by the Boltzmann factor), and resource ratios of the essential elements of life (stoichiometry). Indeed, this bold and visionary proposal is likely to inspire ecologists and provoke much discussion. My goal in assembling this Forum was to work toward a balanced discussion of the power and logic of the metabolic theory of ecology. I have asked both junior and senior scientists to evaluate the ideas presented in the metabolic theory and to go beyond the listing of strong and weak points. As such, this collection of commentaries should be viewed neither as a celebration of the theory nor as a roast of Jim Brown. It should, however, serve as a springboard for future research and refinements of the metabolic theory.

Several themes and axes of admiration and agitation emerge from the forum. The focus on metabolism, and metabolic rate in particular, is an advance that most agree is the fundamental basis for the processes of acquisition of resources from the environment and, ultimately, survival and reproduction of organisms. The combination of size, temperature, and nutrients has compelling predictive power in explaining life-history traits, population parameters, and even broader-scale ecosystem processes. The key point here is that Brown et al. are making a direct link between factors that affect the functioning of individuals and the complex role that those individuals play in communities and ecosystems. Although what we have before us is a proposal for a unified theory of ‘‘biological processing of energy and materials’’ in ecosystems, Brown et al. embrace the unexplained variation and acknowledge other areas of ecology that may not be subject to metabolic laws.

The commentaries presented in this Forum are unanimous in their admiration of Brown et al.’s broad theoretical proposal and its clear predictions. Yet, points of discussion abound and range widely: What really is the correct exponent? Does the scale at which scaling is applied affect its explanatory power? Are the laws really based on mechanism or phenomena? How does the addition of temperature and resource limitation enhance the power of scaling relationships? And, is scaling up from the metabolic rate and body mass of organisms to population dynamics, community structure, and ecosystem processes possible? This Forum ends with Brown’s response to the commentaries. Although there will be continued debate over the correct exponent, the data at hand from the broadest taxonomic groups support quarter powers. There is general agreement over the issue of scale and the fact that, depending on the scale of interest, metabolic theory may have more or less to offer. Finally, nutrient stoichiometry is the most recent addition to metabolic theory, and all agree that further research and refinement will determine the role for such nutrient ratios in the ecological scaling. The benefits of a metabolic theory of ecology are clear. The authors of this Forum have outlined some of the future challenges, and tomorrow’s questions will evaluate these theses.


Metabolism provides a basis for using first principles of physics, chemistry, and biology to link the biology of individual organisms to the ecology of populations, communities, and ecosystems. Metabolic rate, the rate at which organisms take up, transform, and expend energy and materials, is the most fundamental biological rate. We have developed a quantitative theory for how metabolic rate varies with body size and temperature. Metabolic theory predicts how metabolic rate, by setting the rates of resource uptake from the environment and resource allocation to survival, growth, and reproduction, controls ecological processes at all levels of organization from individuals to the biosphere. Examples include:

(1) life history attributes, including development rate, mortality rate, age at maturity, life span, and population growth rate;

(2) population interactions, including carrying capacity, rates of competition and predation, and patterns of species diversity;

(3) ecosystem processes, including rates of biomass production and respiration and patterns of trophic dynamics.

Data compiled from the ecological literature strongly support the theoretical predictions. Eventually, metabolic theory may provide a conceptual foundation for much of ecology, just as genetic theory provides a foundation for much of evolutionary biology.



Key Terms

  • Power Laws
  • Multi-scale
  • Fractals
  • Allometric Scaling Laws
  • Kleiber Law
  • Metabolic Ecology
  • Zipf Distribution
  • allometry
  • biogeochemical cycles
  • body size
  • development
  • ecological interactions
  • ecological theory
  • metabolism
  • population growth
  • production
  • stoichiometry
  • temperature
  • trophic dynamics




Key Sources of Research:


The Origin of Universal Scaling Laws in Biology

Geoffrey B. West



Life’s Universal Scaling Laws

Geoffrey B. West and James H. Brown’sUniversalScalingLaws.pdf



A General Model for the Origin of Allometric Scaling Laws in Biology

Geoffrey B. West, James H. Brown, Brian J. Enquist



Power Laws in Economics: An Introduction

Xavier Gabaix




The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization

Geoffrey B. West, James H. Brown




A general model for ontogenetic growth

Geoffrey B. West, James H. Brown & Brian J. Enquist




Plants on a different scale

Lars O. Hedin




The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms

Geoffrey B. West, James H. Brown, Brian J. Enquist










Complexity and Transdisciplinarity; Science for the 21st Century(?)!





Scaling Laws in Complex Systems




A General Model for the Origin of Allometric Scaling Laws in Biology

Geoffrey B. West, James H. Brown,* Brian J. Enquist




Effects of Size and Temperature on Metabolic Rate

James F. Gillooly,1* James H. Brown,1,2 Geoffrey B. West,2,3 Van M. Savage,2,3 Eric L. Charnov




Growth, innovation, scaling, and the pace of life in cities

Luís M. A. Bettencourt, Jose ́ Lobo, Dirk Helbing, Christian Kuhnert, and Geoffrey B. West




Urban Scaling and Its Deviations: Revealing the Structure of Wealth, Innovation and Crime across Cities

Lu ́ıs M. A. Bettencourt1,2*, Jose ́ Lobo3, Deborah Strumsky4, Geoffrey B. West1,2





Francisco J. Martínez




Allometric Scaling Laws and the Derivation of the Scaling Exponent

Marcel Grunert



Cities, Markets, and Growth: The Emergence of Zipf’s Law

Jeremiah Dittmar

August 10, 2011



Self-similarity and power laws



The fractal nature of nature: power laws, ecological complexity and biodiversity

James H. Brown1,2*, Vijay K. Gupta3, Bai-Lian Li1, Bruce T. Milne1, Carla Restrepo1 and Geoffrey B. West



Metabolic Rate and Kleiber’s Law



Patterns in Nature



Zipf, Power-laws, and Pareto – a ranking tutorial

Lada A. Adamic



The Power of Power Laws



Re-examination of the 3/4-law of Metabolism




Fifth dimension of life and the 4/5 allometric scaling law for human brain

Ji-Huan He, Juan Zhang



Lack of Evidence for 3/4 Scaling of Metabolism in Terrestrial Plants

Hai-Tao LI1*, Xing-Guo HAN2 and Jian-Guo WU



Is West, Brown and Enquist’s model of allometric scaling mathematically correct and biologically relevant?




Evidence against universal metabolic allometry.

Folmer Bokma



An evaluation of two controversial metabolic theories of ecology




Karl J. Niklas1, and Sean T. Hammond



􏱂􏱅Network Allometry



A critical understanding of the fractal model of metabolic scaling

José Guilherme Chaui-Berlinck




Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals

Geoffrey B. West*†‡, William H. Woodruff*§, and James H. Brown

􏰻􏰽􏱃􏱄􏱁􏱂􏰾 􏰼􏰿􏰿􏱁􏱀􏰽􏱃􏱂􏱅
􏱝􏱏 􏱨􏰼􏱂􏱹􏱃􏰼􏱻􏱍􏱑 􏱫􏱏 􏱫􏱹􏱷􏱁􏱻􏱍􏱒 􏱦􏱏􏱞􏱏 􏱞􏰼􏱻􏰼􏲀􏰼􏱂􏱍􏱓 􏱝􏱏 􏱫

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




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.




The Algorithmic Beauty of Plants

Prusinkiewicz, Przemyslaw, Lindenmayer, Aristid



The Algorithmic Beauty of Seashells

Meinhardt H, Prusinkiewicz P, Fowler D


(Springer, New York), 3rd Ed.



The Algorithmic Beauty of Seaweeds, Sponges and Corals

Kaandorp, Jaap A., Kübler, Janet E.



Mathematical Biology

Murray JD




Models of biological pattern formation

Meinhardt H




The chemical basis of morphogenesis.

Turing A




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




Models of biological pattern formation in space and time

Hans Meinhardt




Models of biological pattern formation

Hans Meinhardt,




Cellular Automata, PDEs, and Pattern Formation



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

By Gary William Flake



The Curves of Life

Cook, T


Dover Publications, Inc. New York.



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

Shigeru Kondo1* and Takashi Miura




The Hegemony of Molecular Biology




Modeling seashells


Deborah R. Fowlery􏰣, Hans Meinhardtz and Przemyslaw Prusinkiewicz



The neural origins of shell structure and pattern in aquatic mollusks

Alistair Boettigera, Bard Ermentroutb, and George Oster




Mechanical basis of morphogenesis and convergent evolution of spiny seashells

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




The Geometry and Pigmentation of Seashells

S Coombes





Richie Khandelwal

Sahil Sahni



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

Philip Ball



Mollusc Shell Pigmentation: Cellular Automaton Simulations and Evidence for Undecidability





Pattern Formation in Reaction-Diffusion Systems

Masayasu Mimura



The Natural 3D Spiral

Gur Harary and Ayellet Tal



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



Mechanical growth and morphogenesis of seashells

Derek E. Moulton, Alain Goriely and R ́egis Chirat



Scaling of morphogenetic patterns in continuous and discrete models



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

A A Tsonis, JB Elsner, P A Tsonis




A Model for Pattern Formation on the Shells of Molluscs




Impact of Turing’s Work




The possible role of reaction–diffusion in leaf shape

Nigel R. Franks1* and Nicholas F. Britton



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

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



Turing Patterns

P Ball









Web Resource for Algorithmic Botony




Nicoletta Sala




The Geometry of Seashells

Dr S Coombes







Models for the morphogenesis of the molluscan shell



Modeling Seashell Morphology



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



The Neural Origins of Sea Shell Patterns



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




Cellular Automata as Models of Complexity

Stephen Wolfram,

Nature 311 (5985): 419–424, 1984



Website on Oliva Porphyria



Evolution of patterns on Conus shells

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



Theoretical aspects of pattern formation and neuronal development



20+ Photos Of Geometrical Plants For Symmetry Lovers



Computational models of plant development and form

Przemyslaw Prusinkiewicz and Adam Runions



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

Takashi Miura* and Philip K. Maini



Dynamics of Complex Systems

Yaneer Bar-yam