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Fractal Model of Anomalous Diffusion Eur Biophys J (2015) 44:613–621 DOI 10.1007/s00249-015-1054-5 ORIGINAL PAPER Fractal model of anomalous diffusion Lech Gmachowski1 Received: 15 February 2015 / Revised: 24 April 2015 / Accepted: 9 June 2015 / Published online: 1 July 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract An equation of motion is derived from frac- V0 Mean velocity of the particle or molecule (μm/s) tal analysis of the Brownian particle trajectory in which v0 Mean velocity of the particle or molecule in two the asymptotic fractal dimension of the trajectory has a dimensions (μm/s) required value. The formula makes it possible to calculate 〈r2〉 Mean square displacement of the particle or mol- the time dependence of the mean square displacement for ecule position in two dimensions (μm2) both short and long periods when the molecule diffuses 〈R2〉 Mean square displacement of the particle or mol- anomalously. The anomalous diffusion which occurs after ecule position in three dimensions (μm2) long periods is characterized by two variables, the trans- 〈x2〉 Mean square displacement of the particle or mol- port coefficient and the anomalous diffusion exponent. An ecule position in one dimension (μm2) explicit formula is derived for the transport coefficient, α Anomalous diffusion exponent (–) which is related to the diffusion constant, as dependent on Γ Transport coefficient (μm2/sα) the Brownian step time, and the anomalous diffusion expo- λ Particle or molecule mean free path in two dimen- nent. The model makes it possible to deduce anomalous sions (μm) diffusion properties from experimental data obtained even Λ Particle or molecule mean free path in three dimen- for short time periods and to estimate the transport coef- sions (μm) ficient in systems for which the diffusion behavior has been τ Brownian step time (s) investigated. The results were confirmed for both sub and super-diffusion. Introduction Keywords Contracted or expanded Brownian trajectory · Supported lipid bilayer · Membrane structure · Obstacles to In the interior of biological cells molecules and organelles diffusion · Lipid molecules are immersed in a very crowded aqueous environment which results in specific molecular mobility. Molecule tra- List of symbols jectories in cells are described by competing motion mod- D Diffusion coefficient (μm2/s) els which give the time dependences of the mean square of Dw Differential fractal dimension (–) particle displacement, including free diffusion, anomalous L Trajectory length (μm) diffusion, confined diffusion, and flow or directed motion s Scale of observation (μm) which may result from molecular motor-driven transport. t Time (s) The three diffusive models can occur with directed motion, yielding more complex motion modeled by linear combina- tions of the dependences (Saxton and Jacobson 1997). The * Lech Gmachowski motion of a molecule can be classified by using a method [email protected] based on Bayesian inference to calculate the a-posteriori 1 Institute of Chemistry, Warsaw University of Technology, probability of an observed trajectory on the basis of one of 09‑400 Plock, Poland the models (Monnier et al. 2012). One of the mechanisms 1 3 614 Eur Biophys J (2015) 44:613–621 selected may be anomalous diffusion for which a single that Lévy flights generate super-diffusion. However other molecule trajectory, instead of being the random walk of mechanisms, for example fractional Brownian motion, can the fractal dimension D 2, is either stretched (super-dif- also lead to it (Viswanathan et al. 2008). w = fusion, Dw < 2) or contracted (sub-diffusion, Dw > 2) (Ben- A moving particle or molecule follows linear segments. Avraham and Havlin 2000). For D 1 the motion of the For a very short time the particle travels along the same w = molecule is ballistic. segment and its movement can be regarded as ballistic Several stochastic processes lead to anomalous diffu- (Caspi et al. 2002; Kneller 2011; Wu and Libchaber 2000), sion; these include the continuous-time random walk, frac- for which the fractal dimension D 1. The fractal dimen- w = tional Brownian motion, and Lévy flights and walks. The sion then increases to achieve the asymptotic value after a continuous-time random walk is a stochastic jump pro- very long time. Suppose that the movement can be regarded cess in which random times occur between particle jumps as Brownian, along a trajectory for which the fractal dimen- with arbitrary distributions of jump lengths (Burioni et al. sion is two. Spatial restriction in one direction, however, 2014). Fractional Brownian motion is a symmetric Gauss- can retard the motion of the molecule. Wieser et al. (2007) ian process for which the second moment scales as a power showed equal mobility in the longitudinal and transverse of time (Jeon and Metzler 2010). Lévy flight (Viswanathan directions for proteins diffusing in cellular nanotubes with et al. 2008) is a random walk with a step-lengths probabil- saturation of the mean square displacement with time in ity distribution that is heavy-tailed, so the trajectory of the the perpendicular direction. The measured diffusion coef- molecule contains occasional very long steps. In the Lévy ficient in cellular nanotubules is lower than for unrestricted walk the time to make a step is proportional to its length. environment, and can be estimated by considering confined In cell membranes, anomalous diffusion is probably the mobility phenomenon as early-stage Brownian motion result of both obstacles to diffusion and traps with a dis- (Gmachowski 2014). tribution of binding energies or escape times (Saxton and In sub-diffusion the mechanism is different. The mean Jacobson 1997). Several detailed mechanisms were con- square displacement increases with time but not linearly, sidered by Skaug et al. (2011) as the source of observed as observed for ordinary diffusion. This is a common prop- sub-diffusion: obstruction by the membrane skeleton and erty of all anomalous diffusion phenomena. The particle its bound proteins (Ritchie et al. 2003), inclusion or exclu- or molecule asymptotic trajectory is characterized by two sion from lipid domains (Dietrich et al. 2002), binding to variables, the transport coefficient Γ and the anomalous dif- immobile traps (Saxton 2007), or a combination of these fusion exponent α. The mean square displacement of the (Nicolau et al. 2007). particle or molecule position in two dimensions, detected in Sub-diffusion can be regarded as a result of coexistence experiments with long time periods, is: of normal transport, in time periods in which a particle or molecule locally diffuses freely, and no effective transport, 2 when the object is temporarily trapped as a result of geo- r 4Γ tα (1) = metrical complexity and interactions with the environment. The mean square displacement observed may, after smooth- It seems promising to describe the trajectory of a mole- ing, be described by a power-law dependence of time. This cule by use of a modification of the scale-dependent fractal problem has been extensively studied (Burada et al. 2009; dimension method introduced by Takayasu (1982), origi- Condamin et al. 2008; Goychuk et al. 2014; Santamaria nally for describing the transition of the trajectory fractal et al. 2006). Spatial restriction retards the motion of the dimension from unity for the small scale to two for large molecule so the mean square displacement is smaller than scales. In the model proposed in this paper, the asymptotic for an unrestricted environment. The time taken to achieve fractal dimension of the trajectory of a molecule character- a given diffusion distance is longer. Anomalous diffusion izing its long-term motion can be adjusted. has been widely observed in the plasma membrane of bio- logical cells, and has been used to investigate membrane organization. Sub-diffusion has been proposed as an indi- Model cator of macromolecular crowding in the cytoplasm (Weiss et al. 2004). Let us analyze the fractal dimension of the random walk Super-diffusion is faster than normal diffusion. As ana- particle trajectory. The fractal dimension for a trajectory lyzed by Stauffer et al. (2007), super-diffusion is theoreti- in fully developed Brownian motion is 2. If we consider a cally possible in molecularly crowded environments. In random walk whose mean free path is not negligible, the biological systems, it can be the result of cellular trans- trajectory can be characterized by a scale-dependent frac- port processes and is observed if the diffusion is directed tal dimension. Observing on a scale much shorter than by a motor protein (Goychuk et al. 2014). It is believed the mean free path, one finds the trajectory is nearly a 1 3 Eur Biophys J (2015) 44:613–621 615 line (Kneller 2011). Otherwise, the random walk can be r 1 D reduced to the Brownian motion when analyzed on a suf- L(0) = 1 r w 1 kΛ − (6) ficiently large scale (Bujan-Nunˇez1998 ; Matsuura et al. + 1986; Rapaport 1984, 1985; Takayasu 1990). L(0) is the trajectory contour length equal to the sum of The scale (s)-dependent fractal dimension for a random the line segment lengths. The particle or molecule moves walk trajectory, given in a general form for three-dimen- along a segment with a constant velocity V0. So the contour sional space (Bujan-Nunˇez 1998), is: length can be calculated as the product of the time t and the 1 mean velocity of the particle. Hence: Dw(s) 2 (2) = − 1 s/kΛ 1 r r + τ r Dw 1 V t t (7) where k is a proportionality constant, being a fitting term, 1 − = 0 = Λ + kΛ and Λ is the particle mean free path. Accordingly, Dw (s) varies between 1 if s/kΛ 0 and 2 if s/kΛ . The in which the mean velocity of the particle is replaced by the → →∞ larger the scale of observation, the closer is the random mean free path of diffusing particle divided by the charac- motion to Brownian motion.
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