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On the anomalous diffusion characteristics of membrane-bound proteins
Physics Letters A 342 (2005) 439–442 www.elsevier.com/locate/pla
On the anomalous diffusion characteristics of membrane-bound proteins A.M. Reynolds Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK Received 3 March 2005; received in revised form 18 May 2005; accepted 19 May 2005 Available online 13 June 2005 Communicated by C.R. Doering
Abstract It is shown how the transitions from sub-diffusion to super-diffusion and from sub-diffusion to normal diffusion of individual (independent) proteins within plasma membranes [S. Khan et al., Phys. Rev. E 71 (2005) 041915] can be understood within the framework of a Levy-flight model. 2005 Elsevier B.V. All rights reserved. PACS: 05.40.Fb; 05.40.Jc; 87.15.Aa; 87.14.Ee; 87.15.Kg Keywords: Levy-walks; Fractional diffusion; Proteins; Membranes
The motions of individual proteins on cell surfaces can be observed in computer-enhanced optical microscopy with a spatial resolution of down to tens of nanometres and a typical times resolution of tens of milliseconds [1]. The anomalous diffusion characteristics of membrane-bound proteins observed in such single-particle tracking (SPT) experiments is emerging has a key characteristic of cell membranes because it is both a probe of membrane microstructure and because it has major influence upon reaction kinetics within cell membranes. Recently Khan et al.
E-mail address: [email protected] (A.M. Reynolds). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.086
[2] showed how a simple model in which immobilising events are imposed onto otherwise free Brownian diffusion can describe accurately the sub-diffusive transport characteristics of individual proteins within cell membranes observed in SPT experiments. Their analysis of data from SPT experiments also revealed transitions from sub-diffusion to super-diffusion, and from sub-diffusion to normal diffusion. Here, it is shown how such transitions can be understood in terms of the aforementioned sub-diffusive transport mechanism when it is punctuated by occasional Levy-flights, representative of motor attachment or local membrane flow. Model ingredients underlying the prediction of sub- and super-diffusion are well established [3]. The
440
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
(a)
(b) Fig. 1. Time-evolution (in seconds) of the MSD (in pixels, using a scale of 0.205 µm/px) of protein motions within a plasma membrane. Linear regression fits (dashed lines) to the short and long time behaviours are also shown. Histogram of observed incremental displacements in the x-direction in pixels that occur within the 20 second interval between successive observations. The ensemble is taken over the entire measurement time and over all trajectories within the data set. Gaussian (dotted line) and Fox function with parameter 0.76 (solid line) PDFs with equivalent mean and variance are shown for comparison.
novelty of the approach developed here stems from the manner in which these are combined and in the ability of this simple combination to encapsulate the observed transport characteristics of membrane-bound proteins. Fig. 1 shows the first reported example of a transition from sub- to super-diffusive motions of membrane-bound proteins (more precisely, motions of human major histocompatibility complex class II pro-
teins within the plasma membrane, i.e. within flat areas of the external limiting lipid bilayer membrane of cells) [2]. The dataset comprises 81 tracks, each with 14 data points acquired at 20 second intervals. The path lengths cover distances up to about 10 microns and typically span many lipid domains whose size is of the order 100 nm. The orientation of the coordinate system is arbitrary and determined by the alignment of the CCD camera used to capture the images. There is no evidence of a mean drift in any particular direction, and hence there is no average directed motion present. Ballistic transport believed to occur over times no exceeding several nanoseconds [4–6] is not resolved in the SPT experiments. Linear regression fits to the mean squared particle displacements (MSD) data yields a particle-position variance that grows as x 2 ∼ t 0.76 at intermediate times and as x 2 ∼ t 1.54 at long times. Comparable subdiffusive modes characterised by a 0.75 exponent have been reported for the diffusion of small beads in semiflexible polymers [7], actin melts [8] and the cytoskeleton [9]. For motions in the y-direction the transition is from a sub-diffusive mode of transport characterised by y 2 ∼ t 0.53 to a near normal-diffusive mode characterised by y 2 ∼ t 0.99 . Monte Carlo simulations have provided evidence of a crossover from anomalous to normal diffusion at long times [10]. Establishing whether the anomalous diffusion characteristics of proteins actually show a crossover to normal diffusion at large times or remain anomalous at all times remains a key issue to be resolved. Sub-diffusive transport of membrane-bound proteins may arise from a variety of mechanisms including: obstruction by mobile or immobile proteins; transient binding to immobile proteins; confinement by membrane skeletal corrals; binding or obstruction by the extra-cellular matrix; restrictions to motion imposed by lipid domains; and hydrodynamic interactions ([1] and references therein). These mechanisms have proved difficult to isolate, in large part, because some or all of them occur simultaneously and because their relative importance may depend on the protein and the cell type. Here, following Nagle [11] and Khan et al. [2], cell membranes are simply regarded as being random arrays of continuously changing traps. Sub-diffusion arises when the ∞characteristic residence time within the traps, T = 0 dt tw0 (t) → ∞, i.e. when the distributions of residence times have
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
power-law tails, w0 (t) ∼ t0ν /t 1+ν where 0 < ν < 1 [3]. In a similar vein binding onto molecular motors and other modes of directed-transport is taken to occur to over a broad distribution of lengths, l. Super-diffusion ∞can arise when the mean-square length, l 2 = 0 dl l 2 w1 (t) → ∞, i.e. when the distributions of flight-lengths have power-law tails, η w1 (l) ∼ l1 / l 1+η where 0 < η < 2 [3]. Evidence for super-diffusion by such a Levy-flight mechanism has been observed a photobleaching study of the diffusion of dye bound to cylindrical micelles [12]; a system closely akin to that of membrane-bound proteins. The distributions of residence times and flight-lengths may have a directional dependency because they encapsulate, in part, the effects of restrictions to motion imposed by the heterogeneities in the regions of cell membrane observed in the SPT experiments. For simplicity, the motions of membrane-bound proteins are regarded as being comprised of independent onedimensional processes. This assumption facilitates an analysis in terms of standard approaches [3,13]. The power-law forms of the distributions w0 (t) and w1 (l) cannot extend to arbitrarily small values of the residence times and flight-lengths, t and l, but most cease below some scales, t1 and l1 . Let the cumulative distributions associated with 0 < t < t1 and 0 < l < l1 be denoted by A and B, respectively. Trapping-times and walk-lengths will tend not be drawn from the power-law tails of their respective distributions when the number of steps taken is much less than both Nt = (1 − A)−1 and Nl = (1 − B)−1 . In this case the characteristic duration of a step in the simulated motion of a protein, t0 , and the characteristic length of a walk, l0 , are both finite. Transport will therefore be ballistic at short-times (t < t0 ) and normally-diffusive at later times, with a position-variance evolving according to x 2 = l0 N = l0 t/t0 . When Nt N Nl the maximum duration, tm , of a step within a typical sample of N steps is approximately determined by 1/N = ∞ −υ tm dτ w0 (τ ) ∼ tm and so the time elapsed after N t N steps is t = i=1 ti ≈ N ti ∼ N m dτ τ w0 (τ ) ∼ 1−υ ∼ N 1/υ . In this case, transport will be subNtm diffusive with a position-variance evolving according to x 2 = l0 N ∼ t υ and the distribution of particles at intermediate times will be a solution of the fractional diffusion equation [3], which for an initial point-like distribution of particles is a Fox function
1
P (x, t) = √ 2πkt
441
H 2,0 ν 1,2
x 2 (1 − ν/2, ν) , (1) 4kt ν (0, 1), (1/2, 1)
where k is constant (for a definition of the H function see [3]). It is evident from Fig. 1 that the distribution of observed displacements in the x-direction is, indeed, well approximated by a Fox function probability density function. The data is seen to be less well represented by a Gaussian distribution which tends to under-predict the occurrence of smaller incremental step-lengths. The prediction of a Fox function distribution is, of course, model specific. Sub-ballistic superdiffusion characterised by x 2 ∼ t 2+υ−µ arises when µ > υ and N Nt , Nl [3,13]. An example of a transition from sub-diffusion to sub-ballistic super-diffusive mirroring that observed in the SPT experiments and produced by a numerical implementation of the continuous time random walk model with trapping-times and flight-lengths drawn from the aforementioned distributions is illustrated in Fig. 2. It has been illustrated that the observed transitions from sub-diffusion to super-diffusion and from subdiffusion to normal diffusion of membrane-bound proteins can be understood in terms of Brownian motions punctuated by occasional trapping events and by Levy-flights. In line with now well established princi-
Fig. 2. An example of a transition from sub-diffusion to sub-ballistic super-diffusion mirroring that observed in the SPT experiments and produced by a continuous time random walk model. The ballistic, sub-diffusive and super-diffusive growth of the position-variance is indicated.
442
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
ples [3,13] the occurrence of sub-diffusion is associated with the occurrence of a divergent mean waitingtime whilst super-diffusion stems from a divergent mean flight-length. The novelty of the approach presented here lies in the combination of these processes and in the subsequent identification of transitions between transport modes which mimic observations of membrane-bound proteins. The predicted occurrence of these transitions was supported by the results of numerical simulations. More elaborate models may also encapsulate these transitions.
Acknowledgements I thank Sharon Khan for useful discussions regarding the introduction of a finite characteristic waiting time into the formalism of Metzler and Klafter. This work was funded by the BBSRC and the EPSRC through grant No. 204/E17847.
References [1] M.J. Saxton, K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26 (1997) 273. [2] S. Khan, A.M. Reynolds, I.E.G. Morrison, R.J. Cherry, Phys. Rev. E 71 (2005) 041915. [3] R. Metzler, J. Klafter, Phys. Rep. 339 (1996) 1. [4] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [5] W.L.C. Vaz, P.F. Almeida, Biophys. J. 60 (1991) 1553. [6] J.H. Freed, Annu. Rev. Biophys. 23 (1994) 1. [7] F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, S. Leibler, Phys. Rev. Lett. 77 (1996) 4470. [8] I.Y. Wong, M.L. Gardel, D.R. Reichman, E.R. Weeks, M.T. Valentine, A.R. Bausch, D.A. Weitz, Phys. Rev. Lett. 92 (2004) 178101. [9] M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Biophys. J. 87 (2004) 3518. [10] M.J. Saxton, Biophys. J. 70 (1996) A311. [11] J.F. Nagle, Biophys. J. 63 (1992) 336. [12] A. Ott, J.P. Bouchard, D. Langevin, W. Urbach, Phys. Rev. Lett. 65 (1990) 2201. [13] A. Vazquez, O. Sotolongo-Costa, F. Brouers, Physica A 264 (1999) 424.
On the anomalous diffusion characteristics of membrane-bound proteins A.M. Reynolds Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK Received 3 March 2005; received in revised form 18 May 2005; accepted 19 May 2005 Available online 13 June 2005 Communicated by C.R. Doering
Abstract It is shown how the transitions from sub-diffusion to super-diffusion and from sub-diffusion to normal diffusion of individual (independent) proteins within plasma membranes [S. Khan et al., Phys. Rev. E 71 (2005) 041915] can be understood within the framework of a Levy-flight model. 2005 Elsevier B.V. All rights reserved. PACS: 05.40.Fb; 05.40.Jc; 87.15.Aa; 87.14.Ee; 87.15.Kg Keywords: Levy-walks; Fractional diffusion; Proteins; Membranes
The motions of individual proteins on cell surfaces can be observed in computer-enhanced optical microscopy with a spatial resolution of down to tens of nanometres and a typical times resolution of tens of milliseconds [1]. The anomalous diffusion characteristics of membrane-bound proteins observed in such single-particle tracking (SPT) experiments is emerging has a key characteristic of cell membranes because it is both a probe of membrane microstructure and because it has major influence upon reaction kinetics within cell membranes. Recently Khan et al.
E-mail address: [email protected] (A.M. Reynolds). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.086
[2] showed how a simple model in which immobilising events are imposed onto otherwise free Brownian diffusion can describe accurately the sub-diffusive transport characteristics of individual proteins within cell membranes observed in SPT experiments. Their analysis of data from SPT experiments also revealed transitions from sub-diffusion to super-diffusion, and from sub-diffusion to normal diffusion. Here, it is shown how such transitions can be understood in terms of the aforementioned sub-diffusive transport mechanism when it is punctuated by occasional Levy-flights, representative of motor attachment or local membrane flow. Model ingredients underlying the prediction of sub- and super-diffusion are well established [3]. The
440
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
(a)
(b) Fig. 1. Time-evolution (in seconds) of the MSD (in pixels, using a scale of 0.205 µm/px) of protein motions within a plasma membrane. Linear regression fits (dashed lines) to the short and long time behaviours are also shown. Histogram of observed incremental displacements in the x-direction in pixels that occur within the 20 second interval between successive observations. The ensemble is taken over the entire measurement time and over all trajectories within the data set. Gaussian (dotted line) and Fox function with parameter 0.76 (solid line) PDFs with equivalent mean and variance are shown for comparison.
novelty of the approach developed here stems from the manner in which these are combined and in the ability of this simple combination to encapsulate the observed transport characteristics of membrane-bound proteins. Fig. 1 shows the first reported example of a transition from sub- to super-diffusive motions of membrane-bound proteins (more precisely, motions of human major histocompatibility complex class II pro-
teins within the plasma membrane, i.e. within flat areas of the external limiting lipid bilayer membrane of cells) [2]. The dataset comprises 81 tracks, each with 14 data points acquired at 20 second intervals. The path lengths cover distances up to about 10 microns and typically span many lipid domains whose size is of the order 100 nm. The orientation of the coordinate system is arbitrary and determined by the alignment of the CCD camera used to capture the images. There is no evidence of a mean drift in any particular direction, and hence there is no average directed motion present. Ballistic transport believed to occur over times no exceeding several nanoseconds [4–6] is not resolved in the SPT experiments. Linear regression fits to the mean squared particle displacements (MSD) data yields a particle-position variance that grows as x 2 ∼ t 0.76 at intermediate times and as x 2 ∼ t 1.54 at long times. Comparable subdiffusive modes characterised by a 0.75 exponent have been reported for the diffusion of small beads in semiflexible polymers [7], actin melts [8] and the cytoskeleton [9]. For motions in the y-direction the transition is from a sub-diffusive mode of transport characterised by y 2 ∼ t 0.53 to a near normal-diffusive mode characterised by y 2 ∼ t 0.99 . Monte Carlo simulations have provided evidence of a crossover from anomalous to normal diffusion at long times [10]. Establishing whether the anomalous diffusion characteristics of proteins actually show a crossover to normal diffusion at large times or remain anomalous at all times remains a key issue to be resolved. Sub-diffusive transport of membrane-bound proteins may arise from a variety of mechanisms including: obstruction by mobile or immobile proteins; transient binding to immobile proteins; confinement by membrane skeletal corrals; binding or obstruction by the extra-cellular matrix; restrictions to motion imposed by lipid domains; and hydrodynamic interactions ([1] and references therein). These mechanisms have proved difficult to isolate, in large part, because some or all of them occur simultaneously and because their relative importance may depend on the protein and the cell type. Here, following Nagle [11] and Khan et al. [2], cell membranes are simply regarded as being random arrays of continuously changing traps. Sub-diffusion arises when the ∞characteristic residence time within the traps, T = 0 dt tw0 (t) → ∞, i.e. when the distributions of residence times have
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
power-law tails, w0 (t) ∼ t0ν /t 1+ν where 0 < ν < 1 [3]. In a similar vein binding onto molecular motors and other modes of directed-transport is taken to occur to over a broad distribution of lengths, l. Super-diffusion ∞can arise when the mean-square length, l 2 = 0 dl l 2 w1 (t) → ∞, i.e. when the distributions of flight-lengths have power-law tails, η w1 (l) ∼ l1 / l 1+η where 0 < η < 2 [3]. Evidence for super-diffusion by such a Levy-flight mechanism has been observed a photobleaching study of the diffusion of dye bound to cylindrical micelles [12]; a system closely akin to that of membrane-bound proteins. The distributions of residence times and flight-lengths may have a directional dependency because they encapsulate, in part, the effects of restrictions to motion imposed by the heterogeneities in the regions of cell membrane observed in the SPT experiments. For simplicity, the motions of membrane-bound proteins are regarded as being comprised of independent onedimensional processes. This assumption facilitates an analysis in terms of standard approaches [3,13]. The power-law forms of the distributions w0 (t) and w1 (l) cannot extend to arbitrarily small values of the residence times and flight-lengths, t and l, but most cease below some scales, t1 and l1 . Let the cumulative distributions associated with 0 < t < t1 and 0 < l < l1 be denoted by A and B, respectively. Trapping-times and walk-lengths will tend not be drawn from the power-law tails of their respective distributions when the number of steps taken is much less than both Nt = (1 − A)−1 and Nl = (1 − B)−1 . In this case the characteristic duration of a step in the simulated motion of a protein, t0 , and the characteristic length of a walk, l0 , are both finite. Transport will therefore be ballistic at short-times (t < t0 ) and normally-diffusive at later times, with a position-variance evolving according to x 2 = l0 N = l0 t/t0 . When Nt N Nl the maximum duration, tm , of a step within a typical sample of N steps is approximately determined by 1/N = ∞ −υ tm dτ w0 (τ ) ∼ tm and so the time elapsed after N t N steps is t = i=1 ti ≈ N ti ∼ N m dτ τ w0 (τ ) ∼ 1−υ ∼ N 1/υ . In this case, transport will be subNtm diffusive with a position-variance evolving according to x 2 = l0 N ∼ t υ and the distribution of particles at intermediate times will be a solution of the fractional diffusion equation [3], which for an initial point-like distribution of particles is a Fox function
1
P (x, t) = √ 2πkt
441
H 2,0 ν 1,2
x 2 (1 − ν/2, ν) , (1) 4kt ν (0, 1), (1/2, 1)
where k is constant (for a definition of the H function see [3]). It is evident from Fig. 1 that the distribution of observed displacements in the x-direction is, indeed, well approximated by a Fox function probability density function. The data is seen to be less well represented by a Gaussian distribution which tends to under-predict the occurrence of smaller incremental step-lengths. The prediction of a Fox function distribution is, of course, model specific. Sub-ballistic superdiffusion characterised by x 2 ∼ t 2+υ−µ arises when µ > υ and N Nt , Nl [3,13]. An example of a transition from sub-diffusion to sub-ballistic super-diffusive mirroring that observed in the SPT experiments and produced by a numerical implementation of the continuous time random walk model with trapping-times and flight-lengths drawn from the aforementioned distributions is illustrated in Fig. 2. It has been illustrated that the observed transitions from sub-diffusion to super-diffusion and from subdiffusion to normal diffusion of membrane-bound proteins can be understood in terms of Brownian motions punctuated by occasional trapping events and by Levy-flights. In line with now well established princi-
Fig. 2. An example of a transition from sub-diffusion to sub-ballistic super-diffusion mirroring that observed in the SPT experiments and produced by a continuous time random walk model. The ballistic, sub-diffusive and super-diffusive growth of the position-variance is indicated.
442
A.M. Reynolds / Physics Letters A 342 (2005) 439–442
ples [3,13] the occurrence of sub-diffusion is associated with the occurrence of a divergent mean waitingtime whilst super-diffusion stems from a divergent mean flight-length. The novelty of the approach presented here lies in the combination of these processes and in the subsequent identification of transitions between transport modes which mimic observations of membrane-bound proteins. The predicted occurrence of these transitions was supported by the results of numerical simulations. More elaborate models may also encapsulate these transitions.
Acknowledgements I thank Sharon Khan for useful discussions regarding the introduction of a finite characteristic waiting time into the formalism of Metzler and Klafter. This work was funded by the BBSRC and the EPSRC through grant No. 204/E17847.
References [1] M.J. Saxton, K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26 (1997) 273. [2] S. Khan, A.M. Reynolds, I.E.G. Morrison, R.J. Cherry, Phys. Rev. E 71 (2005) 041915. [3] R. Metzler, J. Klafter, Phys. Rep. 339 (1996) 1. [4] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [5] W.L.C. Vaz, P.F. Almeida, Biophys. J. 60 (1991) 1553. [6] J.H. Freed, Annu. Rev. Biophys. 23 (1994) 1. [7] F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, S. Leibler, Phys. Rev. Lett. 77 (1996) 4470. [8] I.Y. Wong, M.L. Gardel, D.R. Reichman, E.R. Weeks, M.T. Valentine, A.R. Bausch, D.A. Weitz, Phys. Rev. Lett. 92 (2004) 178101. [9] M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Biophys. J. 87 (2004) 3518. [10] M.J. Saxton, Biophys. J. 70 (1996) A311. [11] J.F. Nagle, Biophys. J. 63 (1992) 336. [12] A. Ott, J.P. Bouchard, D. Langevin, W. Urbach, Phys. Rev. Lett. 65 (1990) 2201. [13] A. Vazquez, O. Sotolongo-Costa, F. Brouers, Physica A 264 (1999) 424.