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Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 5 6
Confined mobility in biomembranes modeled by early stage Brownian motion
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Lech Gmachowski ⇑ Warsaw University of Technology, Institute of Chemistry, 09-400 Płock, Poland
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a r t i c l e
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i n f o
Article history: Received 7 January 2014 Received in revised form 4 April 2014 Accepted 23 May 2014 Available online xxxx
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Keywords: Geometrically confined membranes Maximum displacement Confined diffusion Mean free path
a b s t r a c t An equation of motion, derived from the fractal analysis of the Brownian particle trajectory, makes it possible to calculate the time dependence of the mean square displacement for early times, before the Einstein formula becomes valid. The diffusion coefficient increases with the distance travelled which can be restricted by the geometrical conditions. The corresponding diffusion coefficient cannot increase further to achieve a value characteristic for unrestricted environment. Explicit formula is derived for confined diffusivity related to the unrestricted one as dependent on the maximum particle mean square displacement possible normalized by the square of its mean free path. The model describes the lipid and protein diffusion in tubular membranes with different radii, originally fitted by the modified Saffman– Delbrück equation, and the lateral mobility of synthetic model peptides for which the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane. Ó 2014 Published by Elsevier Inc.
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1. Introduction
41
The mobility in biological membranes is described by the Brownian motion although the diffusion coefficient is not determined by the Stokes–Einstein equation. The motion of the nanoparticles in biological systems is usually restricted to bounded domains. Theoretical description of the Brownian motion in biological membranes has been given by Saffman and Delbrück [1], who predicted a logarithmic dependence of the protein diffusion coefficient on the inverse of the size of the protein and on the membrane size if restricted. The space restriction demands a more detailed analysis of the motion of nanoparticles. At very short times the motion of a Brownian particle is regarded as ballistic whereas for long times the particle starts to behave according to Einstein‘s theory [2]. At short distances the Einstein formula is still not valid and the mean square displacement of the particle position is lower than would be for fully developed diffusive motion at the same time. The lower mean square displacement corresponds to the lower diffusion coefficient for early stage of Brownian motion. First description of this phenomenon was done by Langevin [3]. A solution to the corresponding equation was given by Uhlenbeck and Ornstein [4] in the form
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⇑ Tel.: +48 24 262 36 91. E-mail address: [email protected]
hx2 i s t ¼ 1 1 exp 2Dt t s
ð1Þ
This is a time dependence of the mean square displacement of the particle position in one dimension hx2i. The solution contains two parameters which are the diffusion coefficient D and the characteristic time s, being the momentum relaxation time, is calculated as the particle mean free path in one dimension divided by the corresponding mean velocity of the particle.
s ¼ kx =v 0x
ð2Þ
Regarding the functional form of the velocity autocorrelation function in respect to the Langevin equation, a fast exponential transition occurs from the ballistic to the diffusive region, in which the time dependence of the mean square displacement scales with the diffusion coefficient and the momentum relaxation time. Instead of an exponential decay, a long-tail proportional to t3/2 is postulated by Vladimirsky and Terletzky [5] and Hinch [6] for Brownian particle. This form of the velocity autocorrelation function has the experimental confirmation given by Kim and Matta [7]. It is also confirmed by the fractal model of the Brownian particle motion discussed in this paper. A moving particle follows the straight-line segments. At a very short time, when the movement can be considered as ballistic, the particle travels along the same segment with the fractal dimension equal to one. At a very long time the movement can be regarded as Brownian, along a trajectory with the fractal dimension equal to two due to evolution of fractal character of particle trajectory. It
http://dx.doi.org/10.1016/j.mbs.2014.05.002 0025-5564/Ó 2014 Published by Elsevier Inc.
Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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Nomenclature a CV D D h t
v0
91 92 93 94 95 96 97 98
100 101 102 103 104 105 106
107 109
110
111 113
hx2i
radius of cylindrical particle (m) normalized velocity autocorrelation function (–) diffusion coefficient (m2/s) diffusion coefficient in restricted space (m2/s) membrane height (m) time (s) mean velocity of the particle in one direction (m/s)
x
gb gm kx
s
seems reasonable to describe the transition between the two using the scale-dependent fractal dimension changing from one for small scale (ballistic) to two for large ones (diffusive), as postulated by Takayasu [8], who considered an isotropic Brownian motion of a particle in three-dimensional space. The scale (s) dependent fractal dimension for a random walk trajectory, given in a general form by ˇ ez [9], reads Bujan-Nun
1 1 þ s=kk
Dw ðsÞ ¼ 2
ð3Þ
where k is a proportionality constant, being a fitting parameter, and k is the particle mean free path. Accordingly, Dw(s) varies between 1 if s=kk ! 0 and 2 if s=kk ! 1. The bigger the scale of observation, the random motion is thus more close to the Brownian motion. The trajectory length depends on the scale of observation according to the fractal formula
one gets the formula describing the mean square displacement of the particle position in one dimension hx2i as dependent on the number of steps t/s 2 1=2
hx i kx
2 1=2
hx i kþ kx
! ¼k
t
s
2 1=2
hx i
¼ kx
t
s
¼ v 0x t
hx i ¼ ð4Þ
t
s
ð14Þ
¼ kv 0x kx t
ð15Þ
Z
D ¼ v 0x kx
dL ¼ L
r
s kk
1 þ kks
ds s
ð5Þ
ð16Þ
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one gets the result described by Gmachowski [10]
we get the agreement with the Einstein formula hx2i = 2Dt for the value of the fitting parameter k = 2. The final form of Eq. (13) reads
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r 1 ¼ Lð0Þ 1 þ kkr
hx2 i1=2 hx2 i1=2 2þ kx kx
115
118 119
120
122 123 124 125
126 128 129
130
132 133 134 135 136
137
Lð0Þ
0
ð6Þ
L(0) is the trajectory contour length equal to the product of the mean velocity of the particle and time v0t. Hence
1 r rs ¼ ¼ 1 þ kkr v 0 t kt
ð7Þ
¼2
150 151 152 153
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t
s
161 162
163 165 166 167
168
ð17Þ 170
With Eqs (2) and (16) the formula can be rearranged to the form
hx2 i hx2 i1=2 þ ¼1 2Dt v 0x t
160
ð18Þ
or after solving the quadratic Eq. (17)
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where the mean velocity of the particle is replaced by the mean free path of diffusing particle divided by the characteristic momentum relaxation time
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2t=s 1 hx2 i ¼1 2Dt t=s
v 0 ¼ k=s
The fractal model of the particle motion was utilized to formulate [11] the aerosol collision kernel. The applicability of the model of the ballistic-diffusive transition seems to be much wilder, especially to describe the diffusion in restricted environments.
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2. Model of confined Brownian movement
183
Before formulation of the restricted diffusion problem, let us check the reliability of the derived formulae. Eq. (18), which is the sum of diffusive and ballistic contributions, describes the behavior of a Brownian particle for short (hx2i1/2 ? v0xt) and long (hx2i ? 2Dt) times. It clearly indicates that the mean square displacement is less than its value for fully developed diffusive motion (hx2i < 2Dt). Huang et al. [12] investigated the full transition from ballistic to diffusive Brownian motion of small particles in water, observing the behavior of a single particle in an optical trap. Experiments conducted make it possible to verify the model
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ð8Þ
The obtained relation reads
r r t ¼k kþ k k s
ð9Þ
pffiffiffiffiffiffiffiffi Then replacing r by hr 2 i, the root of the mean square displacement of the particle position in three dimensions, one gets the formula describing the mean square displacement of the particle position in three dimensions hr2i as dependent on the number of steps t/s 2 1=2
2 1=2
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Substituting in Eq. (10)
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hr2 i ¼ 3hx2 i pffiffiffi k ¼ 3k x
kþ
hr i k
!
hr i k
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!
147
158
Taking the unrestricted diffusion coefficient described by the kinetic theory, as dependent on the Brownian step parameters
Z
146
157
Integrating with Dw(s) given by Eq. (3) LðrÞ
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and for very long times
kk2x
144
ð13Þ
The derived equation describes a smooth crossover from ballistic to diffusive motion of a Brownian particle, which is essential during the initial stage of the particle motion. For very short times the formula takes the form characteristic for ballistic movement
2
d ln LðsÞ ¼ 1 Dw ðsÞ d ln s
mean square displacement of the particle position in one dimension (m2) bulk fluid viscosity (kg m1 s1) membrane viscosity (kg m1 s1) particle mean free path in one dimension (m) particle momentum relaxation time (s)
¼k
t
s
ð10Þ
ð11Þ ð12Þ
176
ð19Þ
Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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1.4
/2Dt; 1/2/v0xt
1.2
following, the model will be developed to describe the confined diffusion with a special account for biomembranes. The diffusion coefficient increases with the distance travelled, tending to the value characteristic for fully developed diffusive motion in which the Einstein relation hx2i = 2Dt holds. The distance, however, can be restricted by the geometrical conditions. The corresponding diffusion coefficient cannot increase further to achieve a value characteristic for unrestricted environment. The diffusion has isotropic character. The measurements performed by Wieser et al. [13] show the equal mobility in longitudinal and transverse directions of proteins diffusing on cellular nanotubes with the saturation of the mean square displacement with time in perpendicular direction. The restriction in one direction should thus define the diffusion coefficient in all directions. The diffusion coefficient in restricted space is usually calculated by the relation analogous to the Einstein one
/2Dt + 1/2 /v0xt
1.0 0.8
1/2 /v0xt
0.6 0.4
/2Dt
0.2 0.0 1.e-7
1.e-6
1.e-5
t [s]
Dhx2 i
Fig. 1. Confirmation of Eq. (18) using the data measured for 2.5 lm silica particle in water by Huang et al. [12]. The sum (r) of diffusive (h) and ballistic (e) contributions is close to one.
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2
discussed. Normalized mean square displacement hx i/2Dt, measured for 2.5 lm silica particle in water can be compared to the value hx2i1/2/v0xt. The time dependences of both are shown in Fig. 1. The sum of them is close to one in the whole interval of time investigated, which confirms validity of Eq. (18). Using Eq. (19) it is easy to show that the corresponding normalized velocity autocorrelation function has the form
201 2
hx2 i CV ¼ ¼ ð1 þ 2t=sÞ3=2 2 2Ds dðt=sÞ d
203
ð20Þ
206
with expected decay proportional to (t/s)3/2 (Vladimirsky and Terletzky [5], Hinch [6] and Kim and Matta [7]). It greatly differs from that calculated in the same way from Eq. (1)
207 209
C V ¼ expðt=sÞ
205
210 211 212 213 214 215 216 217 218
ð21Þ
Both normalized velocity autocorrelation functions are compared in Fig. 2. The derivation of Eq. (17) is solely a result of the fractal analysis of the structure of Brownian trajectory, in which the mean velocity of the particle and the translational friction coefficient are not defined. Besides of the Brownian motion in a fluid, where the mean velocity is the thermal speed and the translational friction coefficient describes the Stokes formula, the model seems thus to be appropriate for Brownian motion in other systems. In the
1.e-1 CV~(t/τ )-3/2
1.e-2 1.e-3
1.e-5 1.e-1
Langevin equation
1.e+0
fractal model
1.e+1
1.e+2
1.e+3
t/τ Fig. 2. Comparison of the normalized velocity autocorrelation function calculated for the fractal model by Eq. (20) (thick line) with that described by Eq. (21) in respect to the Langevin equation (thin line).
ð22Þ
where the mean square displacement of the particle or molecule position in one dimension hx2i is restricted by the geometrical conditions. From Eqs. (22) and (19) one gets
t
s
¼
2Dhx2 i =D
s¼
4D2hx2 i =D
hx2 i
sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi !, ! Dhx2 i1 hx2 i2 hx2 i2 1 1 hx2 i1 Dhx2 i2 hx2 i1
222 223 224 225 226 227 228 229 230 231 232 233 234
235 237 238 239 240
241
244
245
ð24Þ 247
rffiffiffiffiffiffiffiffiffi, rffiffiffiffiffiffiffiffiffi! Dhx2 i 1 hx2 i 1 hx2 i ¼ pffiffiffi 1 þ pffiffiffi 2D 2D D s s 2 2
248 249 250 251 252 253
254
ð25Þ 256
The confined diffusion coefficient related to the unrestricted one can be calculated rearranging Eq. (24) as
257 258
259
ð26Þ 261
The confined diffusion coefficient related to the unrestricted one is a model function of the restricted value of the mean square displacement divided by the double product of the unrestricted diffusion coefficient and the characteristic time 2
221
2
The diffusion coefficient D, valid in unrestricted environment, is possible to be evaluated using two diffusion coefficients in restricted space Dhx2 i , measured for two different restricted values of the mean square displacement hx2i. The characteristic time for the two cases remains unchanged, so equating the right sides of Eq. (24) one gets after rearrangement
D ¼ Dhx2 i1
220
243
or one obtains the dependence
ð1 Dhx2 i =DÞ
219
ð23Þ
ð1 Dhx2 i =DÞ2
Dhx2 i hx i ¼f 2Ds D
1.e+0
1.e-4
1 x2 ¼ 2 t
CV
204
3
ð27Þ
262 263 264 265
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The parameter Ds is the square mean free path travelled in one direction by the diffusing particle during one step of the Brownian motion (see Eqs. (2) and (16)). The range of hx2i/2Ds < 1/2 corresponds to the situation in which the mean square displacement in one direction is less than the square mean free path in one direction, hx2 i < k2x . This region of strong confinement of the space where the Brownian movement occurs, can be termed as ballistic.
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3. Model analysis using experimental data
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Lipid and protein lateral mobility is essential for biological function. To validate the model, the experimental data were analyzed of lateral mobility measured by Domanov et al. [14] for lipids
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Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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and proteins in tubular membranes. The measured diffusion coefficients were fitted [14] with the Saffman–Delbrück equation [1] modified by Daniels and Turner [15], which predicts a logarithmic dependence of the diffusion coefficient with the tube radius divided by the radius of the diffusing object. The diffusion coefficients measured for limiting values of the range of tube radius 10 and 250 nm, where the close agreement of the fit lines with experimental points is observed, were utilized to calculate the unrestricted diffusion coefficient by Eq. (25) after identifying the values of mean displacement hx2i1/2 to that of tube radius. The obtained values were 3.28 lm2/s for lipids and 2.13 lm2/s for proteins, which are almost the same as Dplanar measured [14], equal 3.3 lm2/s, and 2.2 lm2/s, respectively. The limiting values of lipid and protein diffusion coefficients measured for the lowest and highest tube radii were also utilized to determine by Eq. (24) the values of characteristic time (10.5 and 101 ls) and then the model Ds – parameter. The obtained Ds – parameters were 34.3 and 216 nm2, which correspond to the mean free paths travelled in one direction by the diffusing particle during one step of the Brownian motion, equal 5.86 and 14.7 nm. With the use of the calculated values of Ds – parameter it was possible to recalculate the fit equations to the forms expressed by dependence described by Eq. (27). The fit equations for lipid diffusion and protein diffusion in the whole range of tube radius (10–250 nm) are presented in Fig. 3 in the form of two straight lines and compared to the early stage model of Brownian motion, represented by Eq. (26). The straight lines are pretty close to the model line at limiting values of tube radius but an underestimation can be observed in the middle part of the experimental range. In this region, where the used Saffman–Delbrück fit indeed underestimates the measured data (Fig. 2 and 3 of [14]) of about 20%, the short-time model of Brownian motion seems to be a better model approximating the experimental data of diffusion in geometrically confined membranes. The Saffman–Delbrück equation [1], describing the diffusion coefficient of a membrane inclusion in the form
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g h DSD ¼ ln m 4pgm h gb a
280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314
316
kB T
320 321 322 323 324
325
is widely thought to be valid if the length scale, being the ratio of the membrane viscosity times the membrane thickness to the bounding liquid viscosity, much excides the cylindrical inclusion radius, gmh/gb a, so the particle performs the Brownian motion in a large space. In the case gmh/gb a the space is relatively small. Hughes et al. [16] derived
D/D
319
ð28Þ
1.0 0.9 0.8 0.7 0.6 0.5 peptides 0.4 proteins 0.3 0.2 0.1 model curve 0.0 1.e-2 1.e-1 1.e+0 1.e+1
lipids
1.e+2
1.e+3
/2Dτ Fig. 3. The Saffman–Delbrück equation fits of experimental data [14] for lipid and protein diffusion in the whole range of tube radius (broken lines) compared to the early stage model of Brownian motion (solid line). The points (o) represent the lateral mobility of peptides with diffusion coefficient inversely proportional to the radius of diffusing object and to the thickness of membrane [18].
DHPW ¼
kB T 16gb a
ð29Þ
in which the diffusion coefficient is more strongly dependent on the inclusion radius. Eq. (29) has the experimental confirmation for diffusion coefficients of micron-scale liquid domains in giant unilamellar vesicles [17]. The length scale gmh/gb can be replaced by the membrane radius to obtain the formula for diffusion in geometrically confined membranes [1,15] or by the mean displacement of the particle position in one dimension. The hydrodynamic restriction is replaced in this way by the geometrical one
327 328 329 330 331 332 333 334 335 336
337
gm h=gb ! hx2 i1=2
ð30Þ
Substituting in Eq. (28) one gets
339 340
341
D
hx2 i1=2 ¼ ln 4pgm h a kB T
ð31Þ
the formula used to fit experimental data by Domanov et al. [14] after identifying the values of mean displacement hx2i1/2 to that of tube radius. The results of experiments performed by Gambin et al. [18], who determined the lateral mobility of synthetic model peptides reconstituted into bilayers made of nonionic surfactants, indicate that the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane. In the proposed equation
343 344 345 346 347 348 349 350 351 352
353
D¼
kB Tk 4pgm ha
ð32Þ
355
the characteristic length k is introduced to satisfy dimensionality. In the systems investigated by Gambin et al. [18], as analyzed by Ramadurai et al. [19], the membrane lateral mobility would be obstructed by the presence of respectable part of immobile molecules. This would be the reason for confinement of the diffusion space which should be characterized by the length scale described as hx2i1/2 rather than gmh/gb. The values of diffusion coefficient calculated by Eq. (29), however, are about two orders of magnitude higher than that measured, whereas Eq. (28) gives proper result for the smallest peptide. In this context Eq. (31) should be appropriate. The values of Dhx2 i =D, which is the ratio of measured diffusion coefficient to the unrestricted one calculated by Eq. (28), can be computed. The values of the mean square displacement hx2i were calculated by Eq. (31) and then those of characteristic time s were computed by Eq. (24). In this way the mean free path travelled in one direction by the diffusing object during one step of the Brownpffiffiffiffiffiffi ian motion, kx ¼ Ds, were calculated for peptide in the radius range 10.5–18 Å [18]. The obtained results were almost constant being in a narrow range of 11.5 ± 1.1 nm. The results are presented in Fig. 3 as points for which the values of hx2i/2Ds were calculated with the mean value of kx . They are in the range 0.4–4.1 indicating considerable confinement of the mobility in the system investigated [18].
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4. Discussion and conclusions
380
The time dependence of the mean square displacement for early times, before the Einstein formula becomes valid, has been utilized to reformulate the model to describe the mobility in geometrically confined membranes. According to the model elaborated, the confined diffusivity related to the unrestricted one is dependent on the particle mean square displacement in one dimension normalized by the square of its corresponding mean free path.
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Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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Two sets of experimental data have been considered to validate the model. First the lipid and protein lateral mobilities were analyzed, measured by Domanov et al. [14] in a range of tube radius. The measured diffusion coefficients for limiting values of tube radius 10 and 250 nm, computed from the fit formula, were utilized to calculate the planar diffusion coefficient by Eq. (25). The obtained results for lipids and proteins are almost identical to those measured [14]. This is a strong confirmation of the model presented. The Saffman–Delbrück equation can be considered as a good fit to the experimental values of planar and restricted diffusivities [14], since the obtained parameters such as the membrane viscosity and the disk radii modeling the migrating lipids and proteins are close to the literature data. However, some underestimation of measured diffusion coefficient in the middle part of experimental range of the tube radius can be observed [14]. This underestimation has an explicit reflection in the mutual locations of the Saffman–Delbrück lines and the model curve in Fig. 3. The description of Brownian motion in geometrically restricted biological membranes, predicting a logarithmic dependence of the diffusion coefficient on the tube size and the inverse of the size of diffusing object, although gives a relatively good fit to the experimental data, cannot guarantee the expected asymptotic convergence of the diffusion coefficient to the planar one for large tube radii, when the restriction disappears (Fig. 3). Therefore the proposed model of early stage motion of a Brownian particle, in which such a convergence is observed, seems to be a better description of the mobility in geometrically confined membranes. The expected smooth transition from the restricted to the unrestricted mobility sets a basic advantage of the presented model. The system in which the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane [18] was analyzed as spatially restricted due to the probable presence of immobile molecules [19]. The Saffman– Delbrück equation (31), was used to deduce the values of the mean square displacement hx2i as fitting parameters. The calculated values can be described by the model of early stage Brownian motion. All this confirms the universality of the model which describes the confinement of diffusion in terms of Brownian step parameters and the diffusion distance for both the systems restricted by the membrane geometry and confined by the presence of immobile molecules. The model is not connected with a special dependence of diffusion coefficient on the size of diffusing object. The mean free path travelled in one direction by the diffusing molecule during one step of the Brownian motion is a key model parameter. The mean free path can be deduced from the experi-
5
mental observations of single molecules [20]. If not, the corresponding Ds – parameter can be calculated from two values of diffusion coefficient measured for the same object diffusing in a given biomembrane of different confinements. Then it is possible to determine the full range dependence of the confined mobility. Such procedure seems to be simpler and more reliable than that performed with the use of the Saffman–Delbrück equation.
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References
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[1] P.G. Saffman, M. Delbrück, Brownian motion in biological membranes, Proc. Nat. Acad. Sci. USA 72 (1975) 3111. [2] A. Einstein, Investigations on the Theory of the Brownian Movement, Dover Publications, Mineola, 1956. [3] P. Langevin, Sur la théorie du mouvment brownien, Comptes Rendus de l’Académie des Sciences (Paris) 146 (1908) 530. [4] G.E. Uhlenbeck, L.S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36 (1930) 823. [5] V. Vladimirsky, Ya. Terletzky, Hydrodynamical theory of translational Brownian motion, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 15 (1945) 258–263. [6] E.J. Hinch, Application of the Langevin equation to fluid suspensions, J. Fluid Mech. 72 (1975) 499. [7] Y.W. Kim, J.E. Matta, Long-time behavior of the velocity autocorrelation: a measurement, Phys. Rev. Lett. 31 (1973) 208. [8] H. Takayasu, Differential fractal dimension of random walk and its applications to physical systems, J. Phys. Soc. Jpn 51 (1982) 3057. [9] M.C. Bujan-Nunˇez, Scaling behavior of Brownian motion interacting with an external field, Mol. Phys. 94 (1998) 361. [10] L. Gmachowski, Fractal model of the transition from ballistic to diffusive motion of a Brownian particle, J. Aerosol Sci. 57 (2013) 194. [11] L. Gmachowski, The aerosol particle collision kernel considering the fractal model of particle motion, J. Aerosol Sci. 59 (2013) 47. [12] R. Huang, I. Chavez, K.M. Taute, B. Lukic´, S. Jeney, M.G. Raizen, E.-L. Florin, Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid, Nat. Phys. (2011) 1–5, http://dx.doi.org/10.1038/ NPHYS1953. [13] S. Wieser, G.J. Schütz, M.E. Cooper, H. Stockinger, Single molecule diffusion analysis on cellular nanotubules: implications on plasma membrane structure below the diffraction limit, Appl. Phys. Lett. 91 (2007) 233901. [14] Y.A. Domanov, S. Aimon, G.E.S. Toombes, M. Renner, F. Quemeneur, A. Triller, M.S. Turner, P. Bassereau, Mobility in geometrically confined membranes, Proc. Nat. Acad. Sci. USA 108 (2011) 12605. [15] D.R. Daniels, M.S. Turner, Diffusion on membrane tubes: a highly discriminatory test of the Saffman–Delbrück theory, Langmuir 23 (2007) 6667. [16] B.D. Hughes, B.A. Pailthorpe, L.R. White, The translational and rotational drag on a cylinder moving in a membrane, J. Fluid Mech. 110 (1981) 349. [17] P. Cicuta, S.L. Keller, S.L. Veatch, Diffusion of liquid domains in lipid bilayer membranes, J. Phys. Chem. B 111 (2007) 3329. [18] Y. Gambin, R. Lopez-Esparza, M. Reffay, E. Sierecki, N.S. Gov, M. Genest, R.S. Hodges, W. Urbach, Lateral mobility of proteins in liquid membranes revisited, Proc. Natl. Acad. Sci. USA 103 (2006) 2098. [19] S. Ramadurai, A. Holt, V. Krasnikov, G. van den Bogaart, J.A. Killian, B. Poolman, Lateral diffusion of membrane proteins, J. Am. Chem. Soc. 131 (2009) 12650. [20] S. Wieser, G.J. Schütz, Tracking single molecules in the live cell plasma membrane – Do’s and Don’t’s, Methods 46 (2008) 131.
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Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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No. of Pages 5, Model 5G
9 June 2014 Mathematical Biosciences xxx (2014) xxx–xxx 1
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 5 6
Confined mobility in biomembranes modeled by early stage Brownian motion
3 4 7
Q1
8
Lech Gmachowski ⇑ Warsaw University of Technology, Institute of Chemistry, 09-400 Płock, Poland
10 9 11
a r t i c l e
1 2 3 5 14 15 16 17 18
i n f o
Article history: Received 7 January 2014 Received in revised form 4 April 2014 Accepted 23 May 2014 Available online xxxx
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Keywords: Geometrically confined membranes Maximum displacement Confined diffusion Mean free path
a b s t r a c t An equation of motion, derived from the fractal analysis of the Brownian particle trajectory, makes it possible to calculate the time dependence of the mean square displacement for early times, before the Einstein formula becomes valid. The diffusion coefficient increases with the distance travelled which can be restricted by the geometrical conditions. The corresponding diffusion coefficient cannot increase further to achieve a value characteristic for unrestricted environment. Explicit formula is derived for confined diffusivity related to the unrestricted one as dependent on the maximum particle mean square displacement possible normalized by the square of its mean free path. The model describes the lipid and protein diffusion in tubular membranes with different radii, originally fitted by the modified Saffman– Delbrück equation, and the lateral mobility of synthetic model peptides for which the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane. Ó 2014 Published by Elsevier Inc.
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39 40
1. Introduction
41
The mobility in biological membranes is described by the Brownian motion although the diffusion coefficient is not determined by the Stokes–Einstein equation. The motion of the nanoparticles in biological systems is usually restricted to bounded domains. Theoretical description of the Brownian motion in biological membranes has been given by Saffman and Delbrück [1], who predicted a logarithmic dependence of the protein diffusion coefficient on the inverse of the size of the protein and on the membrane size if restricted. The space restriction demands a more detailed analysis of the motion of nanoparticles. At very short times the motion of a Brownian particle is regarded as ballistic whereas for long times the particle starts to behave according to Einstein‘s theory [2]. At short distances the Einstein formula is still not valid and the mean square displacement of the particle position is lower than would be for fully developed diffusive motion at the same time. The lower mean square displacement corresponds to the lower diffusion coefficient for early stage of Brownian motion. First description of this phenomenon was done by Langevin [3]. A solution to the corresponding equation was given by Uhlenbeck and Ornstein [4] in the form
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62
Q2
⇑ Tel.: +48 24 262 36 91. E-mail address: [email protected]
hx2 i s t ¼ 1 1 exp 2Dt t s
ð1Þ
This is a time dependence of the mean square displacement of the particle position in one dimension hx2i. The solution contains two parameters which are the diffusion coefficient D and the characteristic time s, being the momentum relaxation time, is calculated as the particle mean free path in one dimension divided by the corresponding mean velocity of the particle.
s ¼ kx =v 0x
ð2Þ
Regarding the functional form of the velocity autocorrelation function in respect to the Langevin equation, a fast exponential transition occurs from the ballistic to the diffusive region, in which the time dependence of the mean square displacement scales with the diffusion coefficient and the momentum relaxation time. Instead of an exponential decay, a long-tail proportional to t3/2 is postulated by Vladimirsky and Terletzky [5] and Hinch [6] for Brownian particle. This form of the velocity autocorrelation function has the experimental confirmation given by Kim and Matta [7]. It is also confirmed by the fractal model of the Brownian particle motion discussed in this paper. A moving particle follows the straight-line segments. At a very short time, when the movement can be considered as ballistic, the particle travels along the same segment with the fractal dimension equal to one. At a very long time the movement can be regarded as Brownian, along a trajectory with the fractal dimension equal to two due to evolution of fractal character of particle trajectory. It
http://dx.doi.org/10.1016/j.mbs.2014.05.002 0025-5564/Ó 2014 Published by Elsevier Inc.
Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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Nomenclature a CV D D h t
v0
91 92 93 94 95 96 97 98
100 101 102 103 104 105 106
107 109
110
111 113
hx2i
radius of cylindrical particle (m) normalized velocity autocorrelation function (–) diffusion coefficient (m2/s) diffusion coefficient in restricted space (m2/s) membrane height (m) time (s) mean velocity of the particle in one direction (m/s)
x
gb gm kx
s
seems reasonable to describe the transition between the two using the scale-dependent fractal dimension changing from one for small scale (ballistic) to two for large ones (diffusive), as postulated by Takayasu [8], who considered an isotropic Brownian motion of a particle in three-dimensional space. The scale (s) dependent fractal dimension for a random walk trajectory, given in a general form by ˇ ez [9], reads Bujan-Nun
1 1 þ s=kk
Dw ðsÞ ¼ 2
ð3Þ
where k is a proportionality constant, being a fitting parameter, and k is the particle mean free path. Accordingly, Dw(s) varies between 1 if s=kk ! 0 and 2 if s=kk ! 1. The bigger the scale of observation, the random motion is thus more close to the Brownian motion. The trajectory length depends on the scale of observation according to the fractal formula
one gets the formula describing the mean square displacement of the particle position in one dimension hx2i as dependent on the number of steps t/s 2 1=2
hx i kx
2 1=2
hx i kþ kx
! ¼k
t
s
2 1=2
hx i
¼ kx
t
s
¼ v 0x t
hx i ¼ ð4Þ
t
s
ð14Þ
¼ kv 0x kx t
ð15Þ
Z
D ¼ v 0x kx
dL ¼ L
r
s kk
1 þ kks
ds s
ð5Þ
ð16Þ
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one gets the result described by Gmachowski [10]
we get the agreement with the Einstein formula hx2i = 2Dt for the value of the fitting parameter k = 2. The final form of Eq. (13) reads
117
r 1 ¼ Lð0Þ 1 þ kkr
hx2 i1=2 hx2 i1=2 2þ kx kx
115
118 119
120
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126 128 129
130
132 133 134 135 136
137
Lð0Þ
0
ð6Þ
L(0) is the trajectory contour length equal to the product of the mean velocity of the particle and time v0t. Hence
1 r rs ¼ ¼ 1 þ kkr v 0 t kt
ð7Þ
¼2
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t
s
161 162
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168
ð17Þ 170
With Eqs (2) and (16) the formula can be rearranged to the form
hx2 i hx2 i1=2 þ ¼1 2Dt v 0x t
160
ð18Þ
or after solving the quadratic Eq. (17)
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where the mean velocity of the particle is replaced by the mean free path of diffusing particle divided by the characteristic momentum relaxation time
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2t=s 1 hx2 i ¼1 2Dt t=s
v 0 ¼ k=s
The fractal model of the particle motion was utilized to formulate [11] the aerosol collision kernel. The applicability of the model of the ballistic-diffusive transition seems to be much wilder, especially to describe the diffusion in restricted environments.
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2. Model of confined Brownian movement
183
Before formulation of the restricted diffusion problem, let us check the reliability of the derived formulae. Eq. (18), which is the sum of diffusive and ballistic contributions, describes the behavior of a Brownian particle for short (hx2i1/2 ? v0xt) and long (hx2i ? 2Dt) times. It clearly indicates that the mean square displacement is less than its value for fully developed diffusive motion (hx2i < 2Dt). Huang et al. [12] investigated the full transition from ballistic to diffusive Brownian motion of small particles in water, observing the behavior of a single particle in an optical trap. Experiments conducted make it possible to verify the model
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ð8Þ
The obtained relation reads
r r t ¼k kþ k k s
ð9Þ
pffiffiffiffiffiffiffiffi Then replacing r by hr 2 i, the root of the mean square displacement of the particle position in three dimensions, one gets the formula describing the mean square displacement of the particle position in three dimensions hr2i as dependent on the number of steps t/s 2 1=2
2 1=2
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Substituting in Eq. (10)
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hr2 i ¼ 3hx2 i pffiffiffi k ¼ 3k x
kþ
hr i k
!
hr i k
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!
147
158
Taking the unrestricted diffusion coefficient described by the kinetic theory, as dependent on the Brownian step parameters
Z
146
157
Integrating with Dw(s) given by Eq. (3) LðrÞ
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and for very long times
kk2x
144
ð13Þ
The derived equation describes a smooth crossover from ballistic to diffusive motion of a Brownian particle, which is essential during the initial stage of the particle motion. For very short times the formula takes the form characteristic for ballistic movement
2
d ln LðsÞ ¼ 1 Dw ðsÞ d ln s
mean square displacement of the particle position in one dimension (m2) bulk fluid viscosity (kg m1 s1) membrane viscosity (kg m1 s1) particle mean free path in one dimension (m) particle momentum relaxation time (s)
¼k
t
s
ð10Þ
ð11Þ ð12Þ
176
ð19Þ
Please cite this article in press as: L. Gmachowski, Confined mobility in biomembranes modeled by early stage Brownian motion, Math. Biosci. (2014), http://dx.doi.org/10.1016/j.mbs.2014.05.002
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1.4
/2Dt; 1/2/v0xt
1.2
following, the model will be developed to describe the confined diffusion with a special account for biomembranes. The diffusion coefficient increases with the distance travelled, tending to the value characteristic for fully developed diffusive motion in which the Einstein relation hx2i = 2Dt holds. The distance, however, can be restricted by the geometrical conditions. The corresponding diffusion coefficient cannot increase further to achieve a value characteristic for unrestricted environment. The diffusion has isotropic character. The measurements performed by Wieser et al. [13] show the equal mobility in longitudinal and transverse directions of proteins diffusing on cellular nanotubes with the saturation of the mean square displacement with time in perpendicular direction. The restriction in one direction should thus define the diffusion coefficient in all directions. The diffusion coefficient in restricted space is usually calculated by the relation analogous to the Einstein one
/2Dt + 1/2 /v0xt
1.0 0.8
1/2 /v0xt
0.6 0.4
/2Dt
0.2 0.0 1.e-7
1.e-6
1.e-5
t [s]
Dhx2 i
Fig. 1. Confirmation of Eq. (18) using the data measured for 2.5 lm silica particle in water by Huang et al. [12]. The sum (r) of diffusive (h) and ballistic (e) contributions is close to one.
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2
discussed. Normalized mean square displacement hx i/2Dt, measured for 2.5 lm silica particle in water can be compared to the value hx2i1/2/v0xt. The time dependences of both are shown in Fig. 1. The sum of them is close to one in the whole interval of time investigated, which confirms validity of Eq. (18). Using Eq. (19) it is easy to show that the corresponding normalized velocity autocorrelation function has the form
201 2
hx2 i CV ¼ ¼ ð1 þ 2t=sÞ3=2 2 2Ds dðt=sÞ d
203
ð20Þ
206
with expected decay proportional to (t/s)3/2 (Vladimirsky and Terletzky [5], Hinch [6] and Kim and Matta [7]). It greatly differs from that calculated in the same way from Eq. (1)
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C V ¼ expðt=sÞ
205
210 211 212 213 214 215 216 217 218
ð21Þ
Both normalized velocity autocorrelation functions are compared in Fig. 2. The derivation of Eq. (17) is solely a result of the fractal analysis of the structure of Brownian trajectory, in which the mean velocity of the particle and the translational friction coefficient are not defined. Besides of the Brownian motion in a fluid, where the mean velocity is the thermal speed and the translational friction coefficient describes the Stokes formula, the model seems thus to be appropriate for Brownian motion in other systems. In the
1.e-1 CV~(t/τ )-3/2
1.e-2 1.e-3
1.e-5 1.e-1
Langevin equation
1.e+0
fractal model
1.e+1
1.e+2
1.e+3
t/τ Fig. 2. Comparison of the normalized velocity autocorrelation function calculated for the fractal model by Eq. (20) (thick line) with that described by Eq. (21) in respect to the Langevin equation (thin line).
ð22Þ
where the mean square displacement of the particle or molecule position in one dimension hx2i is restricted by the geometrical conditions. From Eqs. (22) and (19) one gets
t
s
¼
2Dhx2 i =D
s¼
4D2hx2 i =D
hx2 i
sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi !, ! Dhx2 i1 hx2 i2 hx2 i2 1 1 hx2 i1 Dhx2 i2 hx2 i1
222 223 224 225 226 227 228 229 230 231 232 233 234
235 237 238 239 240
241
244
245
ð24Þ 247
rffiffiffiffiffiffiffiffiffi, rffiffiffiffiffiffiffiffiffi! Dhx2 i 1 hx2 i 1 hx2 i ¼ pffiffiffi 1 þ pffiffiffi 2D 2D D s s 2 2
248 249 250 251 252 253
254
ð25Þ 256
The confined diffusion coefficient related to the unrestricted one can be calculated rearranging Eq. (24) as
257 258
259
ð26Þ 261
The confined diffusion coefficient related to the unrestricted one is a model function of the restricted value of the mean square displacement divided by the double product of the unrestricted diffusion coefficient and the characteristic time 2
221
2
The diffusion coefficient D, valid in unrestricted environment, is possible to be evaluated using two diffusion coefficients in restricted space Dhx2 i , measured for two different restricted values of the mean square displacement hx2i. The characteristic time for the two cases remains unchanged, so equating the right sides of Eq. (24) one gets after rearrangement
D ¼ Dhx2 i1
220
243
or one obtains the dependence
ð1 Dhx2 i =DÞ
219
ð23Þ
ð1 Dhx2 i =DÞ2
Dhx2 i hx i ¼f 2Ds D
1.e+0
1.e-4
1 x2 ¼ 2 t
CV
204
3
ð27Þ
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The parameter Ds is the square mean free path travelled in one direction by the diffusing particle during one step of the Brownian motion (see Eqs. (2) and (16)). The range of hx2i/2Ds < 1/2 corresponds to the situation in which the mean square displacement in one direction is less than the square mean free path in one direction, hx2 i < k2x . This region of strong confinement of the space where the Brownian movement occurs, can be termed as ballistic.
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3. Model analysis using experimental data
276
Lipid and protein lateral mobility is essential for biological function. To validate the model, the experimental data were analyzed of lateral mobility measured by Domanov et al. [14] for lipids
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and proteins in tubular membranes. The measured diffusion coefficients were fitted [14] with the Saffman–Delbrück equation [1] modified by Daniels and Turner [15], which predicts a logarithmic dependence of the diffusion coefficient with the tube radius divided by the radius of the diffusing object. The diffusion coefficients measured for limiting values of the range of tube radius 10 and 250 nm, where the close agreement of the fit lines with experimental points is observed, were utilized to calculate the unrestricted diffusion coefficient by Eq. (25) after identifying the values of mean displacement hx2i1/2 to that of tube radius. The obtained values were 3.28 lm2/s for lipids and 2.13 lm2/s for proteins, which are almost the same as Dplanar measured [14], equal 3.3 lm2/s, and 2.2 lm2/s, respectively. The limiting values of lipid and protein diffusion coefficients measured for the lowest and highest tube radii were also utilized to determine by Eq. (24) the values of characteristic time (10.5 and 101 ls) and then the model Ds – parameter. The obtained Ds – parameters were 34.3 and 216 nm2, which correspond to the mean free paths travelled in one direction by the diffusing particle during one step of the Brownian motion, equal 5.86 and 14.7 nm. With the use of the calculated values of Ds – parameter it was possible to recalculate the fit equations to the forms expressed by dependence described by Eq. (27). The fit equations for lipid diffusion and protein diffusion in the whole range of tube radius (10–250 nm) are presented in Fig. 3 in the form of two straight lines and compared to the early stage model of Brownian motion, represented by Eq. (26). The straight lines are pretty close to the model line at limiting values of tube radius but an underestimation can be observed in the middle part of the experimental range. In this region, where the used Saffman–Delbrück fit indeed underestimates the measured data (Fig. 2 and 3 of [14]) of about 20%, the short-time model of Brownian motion seems to be a better model approximating the experimental data of diffusion in geometrically confined membranes. The Saffman–Delbrück equation [1], describing the diffusion coefficient of a membrane inclusion in the form
318
g h DSD ¼ ln m 4pgm h gb a
280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314
316
kB T
320 321 322 323 324
325
is widely thought to be valid if the length scale, being the ratio of the membrane viscosity times the membrane thickness to the bounding liquid viscosity, much excides the cylindrical inclusion radius, gmh/gb a, so the particle performs the Brownian motion in a large space. In the case gmh/gb a the space is relatively small. Hughes et al. [16] derived
D/D
319
ð28Þ
1.0 0.9 0.8 0.7 0.6 0.5 peptides 0.4 proteins 0.3 0.2 0.1 model curve 0.0 1.e-2 1.e-1 1.e+0 1.e+1
lipids
1.e+2
1.e+3
/2Dτ Fig. 3. The Saffman–Delbrück equation fits of experimental data [14] for lipid and protein diffusion in the whole range of tube radius (broken lines) compared to the early stage model of Brownian motion (solid line). The points (o) represent the lateral mobility of peptides with diffusion coefficient inversely proportional to the radius of diffusing object and to the thickness of membrane [18].
DHPW ¼
kB T 16gb a
ð29Þ
in which the diffusion coefficient is more strongly dependent on the inclusion radius. Eq. (29) has the experimental confirmation for diffusion coefficients of micron-scale liquid domains in giant unilamellar vesicles [17]. The length scale gmh/gb can be replaced by the membrane radius to obtain the formula for diffusion in geometrically confined membranes [1,15] or by the mean displacement of the particle position in one dimension. The hydrodynamic restriction is replaced in this way by the geometrical one
327 328 329 330 331 332 333 334 335 336
337
gm h=gb ! hx2 i1=2
ð30Þ
Substituting in Eq. (28) one gets
339 340
341
D
hx2 i1=2 ¼ ln 4pgm h a kB T
ð31Þ
the formula used to fit experimental data by Domanov et al. [14] after identifying the values of mean displacement hx2i1/2 to that of tube radius. The results of experiments performed by Gambin et al. [18], who determined the lateral mobility of synthetic model peptides reconstituted into bilayers made of nonionic surfactants, indicate that the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane. In the proposed equation
343 344 345 346 347 348 349 350 351 352
353
D¼
kB Tk 4pgm ha
ð32Þ
355
the characteristic length k is introduced to satisfy dimensionality. In the systems investigated by Gambin et al. [18], as analyzed by Ramadurai et al. [19], the membrane lateral mobility would be obstructed by the presence of respectable part of immobile molecules. This would be the reason for confinement of the diffusion space which should be characterized by the length scale described as hx2i1/2 rather than gmh/gb. The values of diffusion coefficient calculated by Eq. (29), however, are about two orders of magnitude higher than that measured, whereas Eq. (28) gives proper result for the smallest peptide. In this context Eq. (31) should be appropriate. The values of Dhx2 i =D, which is the ratio of measured diffusion coefficient to the unrestricted one calculated by Eq. (28), can be computed. The values of the mean square displacement hx2i were calculated by Eq. (31) and then those of characteristic time s were computed by Eq. (24). In this way the mean free path travelled in one direction by the diffusing object during one step of the Brownpffiffiffiffiffiffi ian motion, kx ¼ Ds, were calculated for peptide in the radius range 10.5–18 Å [18]. The obtained results were almost constant being in a narrow range of 11.5 ± 1.1 nm. The results are presented in Fig. 3 as points for which the values of hx2i/2Ds were calculated with the mean value of kx . They are in the range 0.4–4.1 indicating considerable confinement of the mobility in the system investigated [18].
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4. Discussion and conclusions
380
The time dependence of the mean square displacement for early times, before the Einstein formula becomes valid, has been utilized to reformulate the model to describe the mobility in geometrically confined membranes. According to the model elaborated, the confined diffusivity related to the unrestricted one is dependent on the particle mean square displacement in one dimension normalized by the square of its corresponding mean free path.
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Two sets of experimental data have been considered to validate the model. First the lipid and protein lateral mobilities were analyzed, measured by Domanov et al. [14] in a range of tube radius. The measured diffusion coefficients for limiting values of tube radius 10 and 250 nm, computed from the fit formula, were utilized to calculate the planar diffusion coefficient by Eq. (25). The obtained results for lipids and proteins are almost identical to those measured [14]. This is a strong confirmation of the model presented. The Saffman–Delbrück equation can be considered as a good fit to the experimental values of planar and restricted diffusivities [14], since the obtained parameters such as the membrane viscosity and the disk radii modeling the migrating lipids and proteins are close to the literature data. However, some underestimation of measured diffusion coefficient in the middle part of experimental range of the tube radius can be observed [14]. This underestimation has an explicit reflection in the mutual locations of the Saffman–Delbrück lines and the model curve in Fig. 3. The description of Brownian motion in geometrically restricted biological membranes, predicting a logarithmic dependence of the diffusion coefficient on the tube size and the inverse of the size of diffusing object, although gives a relatively good fit to the experimental data, cannot guarantee the expected asymptotic convergence of the diffusion coefficient to the planar one for large tube radii, when the restriction disappears (Fig. 3). Therefore the proposed model of early stage motion of a Brownian particle, in which such a convergence is observed, seems to be a better description of the mobility in geometrically confined membranes. The expected smooth transition from the restricted to the unrestricted mobility sets a basic advantage of the presented model. The system in which the diffusion coefficient is inversely proportional to the radius of the diffusing object and to the thickness of the membrane [18] was analyzed as spatially restricted due to the probable presence of immobile molecules [19]. The Saffman– Delbrück equation (31), was used to deduce the values of the mean square displacement hx2i as fitting parameters. The calculated values can be described by the model of early stage Brownian motion. All this confirms the universality of the model which describes the confinement of diffusion in terms of Brownian step parameters and the diffusion distance for both the systems restricted by the membrane geometry and confined by the presence of immobile molecules. The model is not connected with a special dependence of diffusion coefficient on the size of diffusing object. The mean free path travelled in one direction by the diffusing molecule during one step of the Brownian motion is a key model parameter. The mean free path can be deduced from the experi-
5
mental observations of single molecules [20]. If not, the corresponding Ds – parameter can be calculated from two values of diffusion coefficient measured for the same object diffusing in a given biomembrane of different confinements. Then it is possible to determine the full range dependence of the confined mobility. Such procedure seems to be simpler and more reliable than that performed with the use of the Saffman–Delbrück equation.
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