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Fractals in Biology

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

Fractals occur in a wide range of biological applications:

  1. 1)

    In morphology when the shape of an organism (tree) or an organ(vertebrate lung) has a self‐similar brunching structure which can beapproximated by a fractal set (Sect. “Self‐Similar Branching Structures”).

  2. 2)

    Inallometry when theallometric power laws can be deduced from the fractal nature of thecirculatory system (Sect. “ FractalMetabolic Rates”).

  3. 3)

    Inecology whena colony or a habitat acquire fractal shapes due to some SOCprocesses such as diffusion limited aggregation (DLA) or percolation whichdescribes forest fires (Sects. “ Physical Models of BiologicalFractals”–“ Percolation and ForestFires”).

  4. 4)

    Inepidemiology whensome of the features of the epidemics is described by percolation which inturn leads to fractal behavior (Sect. “ Percolation and ForestFires”).

  5. 5)

    In behavioralsciences , when a trajectory of foraging animal...

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Abbreviations

A llometric laws:

Anallometric law describes the relationship between two attributes of livingorganisms y and x, and is usually expressed as a power-law:\( { y\sim x^\alpha } \), where α is thescaling exponent of the law. For example, xcan represent total body mass Mand y can represent the mass ofa brain m b. Inthis case \( { m_{\text{b}}\sim M^{3/4} }\). Another example of an allometric law:\( { B\sim M^{3/4} } \) where B is metabolic rate and M is body mass. Allometric laws can be also found inecology: the number of different species N found in a habitat of area A scales as \( { N\sim A^{1/4} }\).

Radial distribution function:

Radial distribution function g(r) describes howthe average density of points of a set behaves as function ofdistance r from a point of thisset. For an empirical set of N datapoints, the distances between all pair of points are computed and the numberof pairs \( { N_{\text{p}}(r) }\) such that their distance is lessthen r is found. Then\( M(r)=2N_{\text{p}}(r)/N \) gives the averagenumber of the neighbors (mass) of the set withina distance r. For a certaindistance bin \( { r_1<r<r_2 }\), we define \( g[(r_2+r_1)/2]=[M(r_2)-M(r_1)]/[V_d(r_2)-V_d(r_1)] \), where \( V_d(r) =2\pi^{d/2}r^d/[d\Gamma(d/2)] \) is thevolume/area/length of a d‑dimensional sphere/circle/interval ofradius r.

Fractal set:

We definea fractal set with the fractal dimension \( { 0<d_{\text{f}}<d }\) as a set for which\( {M(r)\sim r^{d_{\text{f}}} } \) for\( { r\to\infty } \). Accordingly, for sucha set g(r) decreases as a power law of the distance\( {g(r)\sim r^{-\chi} } \), where\( {\chi=d-d_{\text{f}} }\).

Correlation function:

For a superposition of a fractal set and a set with a finite density defined as\( \rho= \lim_{r\to \infty} M(r)/V_d(r) \),the correlation function is defined as \( { h(r) \equiv g(r)-\rho }\).

Long‐range power law correlations:

The set of points has long-rangepower law correlations (LRPLC) if \( h(r)\sim r^{-\chi} \) for \( { r\to \infty } \) with\( { 0<\chi<d } \). LRPLC indicate the presence of a fractal set with fractaldimension \( { d_{\text{f}} =d-\chi } \) superposed with a uniform set.

Critical point:

Critical point is defined as a point in the systemparameter space (e. g. temperature, \( { T=T_{\text{c}} } \), and pressure \( { P=P_{\text{c}} } \)),near which the system acquires LRPLC

$$h(r)\sim \frac{1}{r^{d-2+\eta}} \exp (r/\xi)\:,$$
(1)

where ξ is the correlation length which diverges near the criticalpoint as \( { \sim |T-T_{\text{c}}|^{-\nu} } \). Here \( { \eta > 0 } \) and \( { \nu > 0 } \) arecritical exponents which depend on the few system characteristics such asdimensionality of space. Accordingly, the system is characterized by fractaldensity fluctuations with \( { d_{\text{f}}=2-\eta } \).

Self‐organized criticality:

Self‐organized criticality (SOC) is a term which describesa system for which the critical behavior characterized by a largecorrelation length is achieved for a wide range of parameters and thusdoes not require special tuning. This usually occurs when a criticalpoint corresponds to an infinite value of a system parameter, such asa ratio of the characteristic time of the stress build up anda characteristic time of the stress release.

Morphogenesis:

Morphogenesis is a branch of developmental biologyconcerned with the shapes of organs and the entire organisms. Several types ofmolecules are particularly important during morphogenesis. Morphogens aresoluble molecules that can diffuse and carry signals that control celldifferentiation decisions in a concentration‐dependent fashion. Morphogenstypically act through binding to specific protein receptors. An importantclass of molecules involved in morphogenesis are transcription factor proteinsthat determine the fate of cells by interacting with DNA. The morphogenesis ofthe branching fractal‐like structures such as lungs involves a dozen ofmorphogenes. The mechanism for keeping self‐similarity of the branches atdifferent levels of branching hierarchy is not yet fully understood. Theexperiments with transgenic mice with certain genes knocked‐out produce micewithout limbs and lungs or without terminal buds.

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  1. Department of Physics, Yeshiva University, New York, USA

    Sergey V. Buldyrev

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Buldyrev, S.V. (2009). Fractals in Biology . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_222

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