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The effect of composition on diffusion of macromolecules in a crowded environment
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Phys. Biol. 12 046003 (http://iopscience.iop.org/1478-3975/12/4/046003) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 198.101.234.89 This content was downloaded on 12/06/2015 at 01:58
Please note that terms and conditions apply.
Phys. Biol. 12 (2015) 046003
doi:10.1088/1478-3975/12/4/046003
PAPER
RECEIVED
30 December 2014 REVISED
15 April 2015 ACCEPTED FOR PUBLICATION
16 April 2015 PUBLISHED
28 May 2015
The effect of composition on diffusion of macromolecules in a crowded environment Svyatoslav Kondrat1,3, Olav Zimmermann2,3, Wolfgang Wiechert1,3 and Eric von Lieres1,3 1 2 3
Forschungszentrum Jülich, IBG-1: Biotechnology, 52425 Jülich, Germany Forschungszentrum Jülich, Jülich Supercomputing Centre, 52425 Jülich, Germany JARA — High-Performance Computing, Germany
E-mail: [email protected] Keywords: macromolecular diffusion, crowding, Brownian dynamics, anomalous subdiffusion, cytoplasm
Abstract We study diffusion of macromolecules in a crowded cytoplasm-like environment, focusing on its dependence on composition and its crossover to the anomalous subdiffusion. The crossover and the diffusion itself depend on both the volume fraction and the relative concentration of macromolecules. In accordance with previous theoretical and experimental studies, diffusion slows down when the volume fraction increases. Contrary to expectations, however, the diffusion is also strongly dependent on the molecular composition. The crossover time decreases and diffusion slows down when the smaller macromolecules start to dominate. Interestingly, diffusion is faster in a cytoplasm-like (more polydisperse) system than it is in a two-component system, at comparable packing fractions, or even when the cytoplasm packing fraction is larger.
1. Introduction The cytoplasm and aqueous compartments of intracellular organelles are crowded with small metabolites and macromolecules, and an extensive research effort has been dedicated to understanding diffusion and reactions in such a crowded environment [1–19]. Crowding profoundly affects intermacromolecular interactions in living systems, thermodynamic activities [20] and diffusion of macromolecules [21–23], and as a consequence also affects the rates of association/disassociation of proteins and enzyme-catalyzed reactions [5–8]—both of which are vital for sustaining the functions of living cells. Previous works have mainly focused on the effects of crowding [21] and intermacromolecular (including hydrodynamic) interactions [22, 23] on diffusion and its anomalous behaviour [9, 24–27]. In particular, experimental studies suggest that the long-time selfdiffusion coefficient of macromolecules in vivo is reduced by roughly one order of magnitude. For instance, the diffusion coefficient of the green fluorescent protein (GFP) in vivo is 10–17 times smaller than in a dilute solution [28, 29]. To understand this diffusion slow-down, McGuffee and Elcock [22] considered a cytoplasm model © 2015 IOP Publishing Ltd
consisting of 50 different types of macromolecules and studied the effect of steric, electrostatic, and van der Waals interactions on diffusion. Using Brownian dynamics (BD) simulations, they showed that steric and electrostatic interactions alone are not sufficient to reproduce the observed slow-down of diffusion. However, the long-time self-diffusion coefficient (DL) turns out to be sensitive to the van der Waals interactions, and it seems possible to match the simulated DL with the diffusion coefficient measured in experiments by varying the depth of the Lennard-Jones potential. Ando and Skolnick [23] argued that the slowdown of diffusion may be due to hydrodynamic interactions, which were not taken into account by [22]. Thus, they showed that accounting for hydrodynamic interactions adequately reproduces the ten-fold reduction of the diffusion coefficient of GFP in the cytoplasm. Another important result of their work is that the details of the macromolecular structure seem to be of negligible importance for BD simulations in a crowded environment, at least for densities up to 350 g l−1. This is shown by comparing the diffusion coefficients in the molecular-shaped system with those in a system where the macromolecules are modeled as spherical particles.
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Living cells are dynamical systems that are maintained by exchanging matter and energy with an environment, and as such are essentially characterized by constant changes of the cell constituents. These changes are due to inflows and outflows of metabolites, their transformation, and production of new proteins and enzymes, which can cause the variation of the packing fraction widely associated with ‘degree of crowding’, and also of the relative concentrations of macromolecules. While it is clear now that the diffusion slows down as a system becomes denser, the effect of macromolecular composition has received almost no attention. The goal of this work is to study the influence of changes in relative concentrations of macromolecules on diffusion, when the volume fraction is kept constant. To accomplish this task, we first perform BD simulations of a simplified two-component system, where the effect of composition can be studied more systematically, and then we compare these results with the results for a model cytoplasm (see figure 1). In both cases we take into account steric, van der Waals, electrostatic, and hydrodynamic interactions, but we model macromolecules as spherical particles for computational efficiency.
2. Model and method Cytoplasm is a rich and diverse environment. For instance, the cytoplasm of E. coli may contain hundreds of different types of macromolecules, and the experimentally available concentrations are often used to investigate the diffusion in crowded environment [21–23]. However, to systematically study the dependence of diffusion on the composition of macromolecules, it is more convenient to first consider a simplified two-component system, and so in this paper we study a system consisting of small and large macromolecules with the hard-core radii σ A = 2.5 nm and σB = 5 nm, which represent tRNA/triphosphate isomerase and RNA polymerase, respectively, and constitute ≈13% and ≈17 % of the volume fraction and c A ≈ 40% and c B ≈ 6% of the number density of E. coli cytoplasm [21, 23]. These two peaks of size distribution are characterized by the ratio of particle concentrations, c A c B = n A n B ≈ 7 (ni is the number of moles). We will use c AB = c A c B = 7 as in some sense representative, and we will compare it with other values of cAB as well as with a realistic cytoplasm model borrowed from [23]. We model macromolecules as spherical LennardJones particles. The Lennard-Jones potential for large spherical molecules [23, 30] was used to describe nonspecific van der Waals interactions. The depth of the potential well was taken as ϵLJ = 0.375 kcal mol−1 in all cases. The electrostatic interactions were calculated using the Debye–Hückel [31] approximation for 2
spheres, with the inverse Debye screening length κ = 0.1 Å−1 and the dielectric constant ε = 78.54. The total charges were q A = −76 and qB = −41 in units of elementary charge, which correspond to tRNA and RNA polymerase, respectively. For such large charges, the Debye–Hückel approximation is not correct, and effective renormalized charges must be used [32]; we used a renormalization factor of 0.2, as in [23]. The hydrodynamic interactions were calculated by the Rotne–Prager–Yamakawa [33, 34] diffusion tensor, with the Ewald summation [35] used to account for the long-range nature of the hydrodynamic forces; the short-range contribution was neglected. It is known that ‘coarse-graining’ of macromolecules tends to overestimate their diffusion coefficients [36]. For this reason, the hydrodynamic radii, which were larger than the hard-core radii, were used to match the bare diffusion coefficients at a given viscosity (ν = 0.01 Poise); in particular, we took R A = 33 nm and R B = 66 nm, which correspond to the bare diffusion coefficients D0, A ≈ 6.58 Å2 ns−1 and D0, B ≈ 3.29 Å2 ns−1, respectively. We recall that a similar approach has been used in other studies [23], showing a satisfactory comparison with atomistic models. BD simulations were performed using a customized BD_BOX code [37, 38]. Particles were packed in a periodic box of size 75 × 75 × 75 nm3. The Ermak–McCammon algorithm [39] with the Iniesta– de la Torre [40] correction was applied to solve the BD equations of motion. The time step was 1 ps or smaller. To gather enough statistics 5 to 10 independent simulations were performed, starting from random initial configurations.
3. Self-diffusion and crossover to anomalous subdiffusion In simulations or experiments, the self-diffusion coefficient can be measured by tracing the particle position (r ) and calculating the mean-square displacement (MSD). Then the Einstein–Helfand relation [41, 42] gives MSD (t ) = 〈 [r (t ) − r (0)]2 〉 = 6Dt α in three dimensions, where α = 1. However, this linear behaviour of MSD at all times occurs only when there is no additional length or time scale in the system, as in the case of the Brownian motion of a single colloidal particle (i.e., in infinite dilution). In a system with two or more characteristic length (or time) scales, the MSD can behave differently, with the exponent α deviating from unity. The anomalous subdiffusion [9, 26, 27] with α < 1 is frequently observed in crowded systems. Let us consider, as an example, the diffusion of macromolecules (or colloids) immersed in an aqueous solution. Apart from momentum relaxation, there is a separation of time scales in MSD (t ) determined by macromolecular collisions. On short time scales, the behaviour of MSD (t ) reflects the viscosity of a solvent,
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Figure 1. Snapshots of BD simulations (a) of a model cytoplasm and (b) of a dense two-component system. The volume fraction occupied by macromolecules is approximately 19% in both cases.
and hence the diffusion coefficient is comparable to the bare diffusion coefficient, D0 = k B T 6πνR , with ν being the viscosity and R being the hydrodynamic radius, as before. At much longer time scales, the macromolecule motion is affected by the collisions with other macromolecules; the diffusion coefficient falls, but the MSD may still behave linearly with time. These two diffusion coefficients are called short- and longtime self-diffusion coefficients, DS and DL, respectively. Another possibility is the anomalous subdiffusion in which molecules are permanently ‘trapped and released’ by their neighbours. In any case, the deviation from the normal short-time diffusion will roughly occur on the time scales determined by macromolecular collisions. The crossover time to anomalous subdiffusion (either transient or permanent) is difficult to calculate analytically, and so we resort to simple estimates. For the ‘free diffusion path’ for a molecule of type k, we write λ k ≈ aρ−1
3
⎛ − ⎜⎜ σk + ⎝
M
⎞
i
⎠
∑x i σi⎟⎟
(1)
where ρ = N V is the total density, x i = Ni N is the concentration, Ni is the number of macromolecules of M type i, N = ∑ Ni is the total number of macroi molecules, and M is the number of different types of macromolecules; σi is the hard-core radius, and a is the proportionality factor, for which we choose a = 1. (Another possible choice is the Wigner–Seitz radius, which gives aWS = (3 4π )1 3. More detailed calculations give a = Γ (4 3) αWS , see [43, 44], which differs by a factor of ≈0.55 from our definition. We choose a = 1 for simplicity, and to get a lower estimate for the time step in point-based BD simulations (see below). Note, however, that this definition may overestimate the maximum time step in molecular-based BD simulations.) In equation (1), the first term is (approximately) the average distance between the molecules, and the second term in the brackets gives the average distance at close contact (assuming that 3
the system is homogeneous). We now use the bare self-diffusion coefficient, D0, i , so for the inter-collision and hence crossover time for a molecule of type i, we get
τi = λ i2 D0, i .
(2)
The crossover time at a constant total packing/volume fraction depends strongly on composition (figure 2). It is larger for smaller molecules, as one might expect, and increases with a decrease in the fraction of bigger macromolecules. The variation of τ with concentration is weaker when the total excluded volume, rather than the packing fraction, is kept constant (lower inset in figure 2). Interestingly, the crossover time is also larger for more polydisperse systems like cytoplasm (upper inset of figure 2). 3.1. Time steps in BD simulations There are two types of BD simulations typically used in the literature. First, there are microscopic or coarsegrained simulations, where the molecular size is taken into account either on the atomistic or the coarsegrained level [21–23]. Second, there are BD simulations where macromolecules are treated as point-like particles [45, 46]. The crossover time sets natural limits for the time steps in these BD simulations (note that there are also other restrictions on the time step, such as intermolecular forces, reaction rates, etc.). Clearly, in microscopic Brownian (and also molecular) dynamic simulations, the time step must be smaller than the crossover time, (i.e., Δt ≪ τ). On the other hand, in point-particle simulations, the time step can be larger than the crossover time (more precisely, larger than the time when the long-time normal diffusion is ‘re-established’). It is important to note that in point-particle BD, the long-time selfdiffusion coefficient must be taken, but it is clearly not possible to take the short-range diffusion coefficient, as there is no way to reproduce the correct long-time behaviour in systems where particles have no size. It may happen that the long-time (t → ∞) behaviour is not reached in certain problems, as for
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Figure 2. Crossover time (τ ) to anomalous subdiffusion for a two-component system and for a cytoplasm model; τ was obtained using equation (2). The hard-core radii σB = 5 nm and σ A = 2.5 nm. The packing fraction was kept fixed, η = (4π 3) ∑i ρi σi3 = 0.186168, where ρi = Ni V is the density of molecules of type i. The upper inset shows the crossover time as a function of the packing fraction for x A = 0.875 and compares it with τ of the same macromolecule in a more realistic cytoplasm model borrowed from [23]. The lower inset shows the crossover time for a fixed excluded volume, Vex(i) = (4π 3) ∑ j ρ j (σi + σ j )3 ≈ 1.5. Notably, the variation of τ is smaller when the excluded volume, rather than the packing fraction, is kept constant. In the lower inset the axis scales and limits are the same as on the main plot.
instance when the crossover length (roughly MSD1 2 (τ)) is comparable to the size of a system. In this case, either microscopic simulations (and the short-time diffusion) can be used, or subdiffusion can be modeled by fractional diffusion [27, 47–50]. However, the latter requires that the exponent, α, of subdiffusion can be obtained, which may not generally be the case [51]. In our BD simulations, we took Δt = 1 ps, and in some cases even smaller values of Δt were used, which is well below the calculated crossover times (see figure 2).
4. Results of BD simulations 4.1. MSD and crossover time: dependence on packing fraction Despite its simplicity, the estimated crossover time, equation (2), agrees surprisingly well with the results of BD simulations. Figure 3 shows the MSD versus time for a two-component system with different packing fractions, where symbols denote the crossover time, τ , calculated by equation (2). τ decreases when the volume fraction increases, and coincides approximately with the time when the MSD starts to deviate from the normal diffusion (i.e., where MSD ∼ t α , α = 1, for t < τ ). Figure 3 also demonstrates that diffusion slows down when the volume fraction increases. This is in accordance with the experiments [28, 29] and other simulations [22, 23]. Note that both the diffusion coefficient and the crossover time decrease. It is not clear at this stage whether the anomalous subdiffusion (α < 1) will persist or the normal diffusion with a smaller diffusion coefficient will re-establish at long 4
Figure 3. MSD as a function of time for a two-component dense system from BD simulations. The MSD is shown for a few values of the volume fraction (100 × η where η = (4π 3) ∑i ρi σi3 with ρi being the density and σi being the radius of a molecule of type i = {A, B}). The hard-core radii are σ A = 2.5 and σB = 5 nm, and the concentration x A = NA N = 0.785. The symbols show the crossover time to anomalous subdiffusion calculated from equation (2). The thin dashed line shows the linear behaviour, MSD ∼ t , corresponding to normal diffusion.
times. Our preliminary results suggest, however, that the latter is more likely the case, with the window of anomalous subdiffusion extending from t = τ to about a few to tens of microseconds. For instance, for the volume fraction 19% (a less computationally demanding case with a relatively small window of transient subdiffusivity), we observe the long-time normal diffusion for t > τL ≈ 4.7 μs as follows from the saturation of MSD/t. For the σ A = 2.5 nm macromolecule, the long-time self-diffusion coefficient is D L, A ≈ 1.04 Å2 ns−1, which is more than sixfold smaller than the diffusion coefficient in infinite dilution (D0, A ≈ 6.58 Å2 ns−1). 4.2. Influence of composition To study the effect of composition, one can keep the number density, ρ = N V , fixed and vary the concentration of different macromolecules, x i = Ni N , where i = {A, B} for a two-component system. However, the density alone does not distinguish between molecular sizes, and thus does not tell us how crowded the system actually is. Therefore, it seems more natural to keep the packing fraction constant, η = (4πρ 3) ∑i x i σi2 = const, when varying the concentrations, xi. To do this consistently, it is convenient to rewrite, for a two-component system, η = (v A V )(NA + r 3NB ), where r = σB σ A and v A = (4π 3) σ A3 is the volume of molecule A. To vary xi at fixed η amounts to varying NB and changing NA according to NA (NB ) = N A(max) − r 3NB , where N A(max) = Vη v A . Clearly, NB can be varied from zero to NB(max) = N A(max) r 3, where NB(max) implies NA = 0. In what follows, we took N A(max) = 1200, which with corresponds to η = 0.186168 ≈ 0.19 V = (75 nm)3 and σ A = 2.5 nm.
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Figure 4. MSD as a function of time for a two-component dense system from BD simulations. The MSD is shown for a few values of concentration of smaller macromolecules (x A ). The volume fraction 100 × η ≈ 19% , where η = (4π 3) ∑i ρi σi3 ≈ 0.19, ρi is the density and σi the radius of a molecule of type i = {A, B}. The macromolecule radii are σ A = 2.5 and σB = 5 nm. The symbols show the crossover time to anomalous subdiffusion calculated from equation (2). The thin dashed line shows the linear behaviour, MSD ∼ t , corresponding to normal diffusion.
Figure 4 shows the influence of composition for a few values of the concentration of smaller macromolecules, x A = NA N . The diffusion clearly slows down when xA increases. This correlates with a decrease of the crossover time with increasing xA (figure 2). By keeping the packing fraction constant and increasing the number of A (smaller) molecules, the average distance between the molecules decreases (see equation (1) and figure 2), and the excluded volume increases. The volume excluded by i molecule to a j molecule is vex(ij) = (4π 3)(σi + σ j )3. For instance, consider an A molecule as a test particle. The volume excluded to it by another A molecule is vexAA = 8v A ; the volume excluded to it by a big (B) molecule is vexAB = 27v A (vA is the volume of the A molecule). Trivial arithmetic shows that replacing one B molecule by 8 A molecules keeps the volume taken up by all molecules constant, but increases the excluded volume by 64 27 ≈ 2.4. It is likely that this increase in the excluded volume and a decrease in the free diffusion length are responsible for the diffusion slow-down when the number of smaller molecules increases in a crowded system. The allusion to this is that the crossover time varies less with the concentration when the excluded volume, rather than the packing fraction, is kept constant (the lower inset in figure 2). Physically, adding a small number of bigger (B) macromolecules mainly creates a few steric obstacles for other molecules, while some distance away from them, the system remains relatively dilute. On the contrary, adding many smaller macromolecules (of the same total volume) makes the system effectively denser and leads to the decrease in diffusion. Surprisingly, the slow-down in diffusion caused by a change in concentration (at the same volume fraction) is comparable or even stronger than the slow5
Figure 5. Comparison of the MSD of a diffusing macromolecule of size 2.5 nm in a model cytoplasm and in a twocomponent system. The diffusion is faster in the cytoplasm although the packing fractions of both systems are comparable. The cytoplasm model is taken from [23] and has a volume fraction of 19.2%. The two-component system consists of only two types of macromolecules and has volume fractions of 18.6% and 11.6%, and the concentration of smaller molecules x A = 0.875.
down induced by an increase of the packing fraction (compare figures 3 and 4). This turns out to be true also for more complex systems. Figure 5 shows that the (middle-time) diffusion in the cytoplasm is faster than in a two-component system at comparable packing fractions. Similarly, the diffusion is comparable in the cytoplasm and in the two-component system, although the cytoplasm is over 50% more crowded than the two-component system.
5. Conclusions We have studied the diffusion of macromolecules in a crowded environment. Using BD simulations, we characterized diffusion by calculating the MSD. At short times, the MSD depends linearly on time, MSD ∼ t α , where α = 1, and we observe normal diffusion. At longer times, the collisions between macromolecules become important, the MSD starts to deviate from the normal behaviour, and the diffusion develops into being anomalously slow (a so-called subdiffusion, α < 1). We have estimated the crossover time to anomalous subdiffusion from the free diffusion length (equations (1) and (2)). In defiance of its simplicity, this estimate agrees surprisingly well with the results of BD simulations (figure 3). In accordance with other theoretical [22, 23] and experimental [28, 29] studies, the diffusion slows down when the volume fraction increases (figure 3). Surprisingly, however, the diffusion and crossover time seem to depend strongly on the molecular composition as well. The diffusion slows down when the number of smaller molecules increases while the volume fraction is kept constant (figure 4). This is because the average distance between the macromolecules becomes shorter, effectively rendering the
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system denser. For the same reason, the crossover time to anomalous diffusion becomes smaller (figure 2). The slow-down of diffusion induced by a change in composition is comparable to or can even be stronger than the slow-down caused by an increase in the volume fraction. A striking example is the cytoplasm model, in which the (middle-time) diffusion turns out to be faster than in a two-component system at comparable volume fractions (figure 5). This study suggests that the volume fraction alone is not a unique measure of how crowded a system is with respect to diffusion. Indeed, diffusion depends sensitively on molecular composition, and can in fact be faster in systems with higher volume fractions. Weiss et al have proposed using an exponent of the anomalous subdiffusion (i.e., α in MSD ∼ t α ) as a measure of crowdedness [9]. A crossover time from the short-time normal diffusion to the anomalous subdiffusion, τ, may serve as an additional parameter characterizing the degree of crowding. It is interesting to note, however, that both τ and α depend on molecular sizes and on the interaction between molecules (the latter is not taken into account in a simple estimate (2)). Thus, the so-defined ‘degree of crowding’ can be different for two systems with the same volume fraction but with different intermolecular interactions. More importantly, it will differ for different molecules in the same system. For instance, as figure 2 suggests, a system can look crowded in the eyes of a large molecule, but can still be relatively dilute for smaller molecules.
Acknowledgments This work was supported by BMBF Grant No. 031A129. The authors acknowledge the computing time granted on supercomputers JUROPA and JUDGE of Jülich Supercomputing Centre (JSC). SK is grateful to Paweł Zieliński and Maciej Długosz for fruitful discussions and support with BD_BOX, and to Pradeep Burla for technical assistance.
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[9] Weiss M, Elsner M, Kartberg,and F and Nilsson T 2004 Biophys. J. 87 3518 [10] Dix J A and Verkman A 2008 Annu. Rev. Biophys. 37 247 [11] Geyer T 2011 BMC Biophys. 4 7 [12] Dlugosz M and Trylska J 2011 BMC Biophys. 4 3 [13] Mereghetti P, Kokh D, McCammon J A and Wade R C 2011 BMC Biophys. 4 2 [14] Schöneberg J and Noé F 2013 PLOS One 8 e74261 [15] Foffi G, Pastore A, Piazza F and Temussi P A 2013 Phys. Biol. 10 040301 [16] Qin S, Mittal J and Zhou H-X 2013 Phys. Biol. 10 045001 [17] Sanfelice D, Politou A, Martin S R, Rios P D L, Temussi P and Pastore A 2013 Phys. Biol. 10 045002 [18] Abriata L A, Spiga E and Peraro M D 2013 Phys. Biol. 10 045006 [19] Fanelli D, McKane A J, Pompili G, Tiribilli B, Vassalli M and Biancalani T 2013 Phys. Biol. 10 045008 [20] Minton A P 1981 Biopolymers 20 2093 [21] Ridgway D, Broderick G, Lopez-Campistrous A, Ruaini M, Winter P, Hamilton M, Boulanger P, Kovalenko A and Ellison M J 2008 Biophys. J. 94 3748 [22] McGuffee S R and Elcock A H 2010 PLoS Comput. Biol. 6 e1000694 [23] Ando T and Skolnick J 2010 Proc. Natl Acad. Sci. USA 107 18457 [24] Minton A P 2001 J. Biol. Chem. 276 10577 [25] Verkman A S 2002 Trends Biochem. Sci. 27 27 [26] Banks D S and Fradin C 2005 Biophys. J. 89 2960 [27] Regner B M, Vučinić D, Domnisoru C, Bartol T M, Hetzer M W, Tartakovsky D M and Sejnowski T J 2013 Biophys. J. 104 1652 [28] Elowitz M B, Surette M G, Wolf P-E, Stock J B and Leibler S 1999 J. Bacteriol. 181 197 (PMID: 9864330) [29] Konopka M C, Shkel I A, Cayley S, Record M T and Weisshaar J C 2006 J. Bacteriol. 188 6115 [30] Henderson D, Duh D-M, Chu X and Wasan D 1997 J. Colloid. Interface. Sci. 185 265 [31] Israelachvili J N 1991 Intermolecular and Surface Forces (New York: Academic) [32] Levin Y 2002 Rep. Prog. Phys. 65 1577 [33] Rotne J and Prager S 1969 J. Chem. Phys. 50 4831 [34] Yamakawa H 1970 J. Chem. Phys. 53 436 [35] Smith E R, Snook I K and van Megen W 1987 Physica A 143 441 [36] Huang D M, Faller R, Do K and Moulè A J 2010 J. Chem. Theory Comput. 6 526 [37] Dlugosz M, Zielinski P and Trylska J 2011 J. Comput. Chem. 32 2734 [38] http://browniandynamics.org/index.html [39] Ermak D L and McCammon J A 1978 J. Chem. Phys. 69 1352 [40] Iniesta A and de la Torre J G 1990 J. Chem. Phys. 92 2015 [41] Frenkel D and Smith B 1996 Understanding Molecular Simulations (New York: Academic) [42] Gubbins K E, Liu Y-C, Moore J D and Palmer J C 2011 Phys. Chem. Chem. Phys. 13 58 [43] Hertz P 1909 Math. Ann. 67 387 [44] Chandrasekhar S 1943 Rev. Mod. Phys. 15 1 [45] Andrews S and Bray D 2004 Phys. Biol. 1 137 [46] Andrews S S, Addy N J, Brent R and Arkin A 2010 PLoS Comput. Biol. 6 e1000705 [47] Metzler R, Barkai E and Klafter J 1999 Phys. Rev. Lett. 82 3563 [48] Ilyin V, Procaccia I and Zagorodny A 2010 Phys. Rev. E 81 030105 [49] Fritsch C C and Langowski J 2012 J. Chem. Phys. 137 064114 [50] Höfling F and Franosch T 2013 Rep. Prog. Phys. 76 046602 [51] Hellmann M, Klafter J, Heermann D W and Weiss M 2011 J. Phys: Condens. Matter 23 234113
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The effect of composition on diffusion of macromolecules in a crowded environment
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Phys. Biol. 12 046003 (http://iopscience.iop.org/1478-3975/12/4/046003) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 198.101.234.89 This content was downloaded on 12/06/2015 at 01:58
Please note that terms and conditions apply.
Phys. Biol. 12 (2015) 046003
doi:10.1088/1478-3975/12/4/046003
PAPER
RECEIVED
30 December 2014 REVISED
15 April 2015 ACCEPTED FOR PUBLICATION
16 April 2015 PUBLISHED
28 May 2015
The effect of composition on diffusion of macromolecules in a crowded environment Svyatoslav Kondrat1,3, Olav Zimmermann2,3, Wolfgang Wiechert1,3 and Eric von Lieres1,3 1 2 3
Forschungszentrum Jülich, IBG-1: Biotechnology, 52425 Jülich, Germany Forschungszentrum Jülich, Jülich Supercomputing Centre, 52425 Jülich, Germany JARA — High-Performance Computing, Germany
E-mail: [email protected] Keywords: macromolecular diffusion, crowding, Brownian dynamics, anomalous subdiffusion, cytoplasm
Abstract We study diffusion of macromolecules in a crowded cytoplasm-like environment, focusing on its dependence on composition and its crossover to the anomalous subdiffusion. The crossover and the diffusion itself depend on both the volume fraction and the relative concentration of macromolecules. In accordance with previous theoretical and experimental studies, diffusion slows down when the volume fraction increases. Contrary to expectations, however, the diffusion is also strongly dependent on the molecular composition. The crossover time decreases and diffusion slows down when the smaller macromolecules start to dominate. Interestingly, diffusion is faster in a cytoplasm-like (more polydisperse) system than it is in a two-component system, at comparable packing fractions, or even when the cytoplasm packing fraction is larger.
1. Introduction The cytoplasm and aqueous compartments of intracellular organelles are crowded with small metabolites and macromolecules, and an extensive research effort has been dedicated to understanding diffusion and reactions in such a crowded environment [1–19]. Crowding profoundly affects intermacromolecular interactions in living systems, thermodynamic activities [20] and diffusion of macromolecules [21–23], and as a consequence also affects the rates of association/disassociation of proteins and enzyme-catalyzed reactions [5–8]—both of which are vital for sustaining the functions of living cells. Previous works have mainly focused on the effects of crowding [21] and intermacromolecular (including hydrodynamic) interactions [22, 23] on diffusion and its anomalous behaviour [9, 24–27]. In particular, experimental studies suggest that the long-time selfdiffusion coefficient of macromolecules in vivo is reduced by roughly one order of magnitude. For instance, the diffusion coefficient of the green fluorescent protein (GFP) in vivo is 10–17 times smaller than in a dilute solution [28, 29]. To understand this diffusion slow-down, McGuffee and Elcock [22] considered a cytoplasm model © 2015 IOP Publishing Ltd
consisting of 50 different types of macromolecules and studied the effect of steric, electrostatic, and van der Waals interactions on diffusion. Using Brownian dynamics (BD) simulations, they showed that steric and electrostatic interactions alone are not sufficient to reproduce the observed slow-down of diffusion. However, the long-time self-diffusion coefficient (DL) turns out to be sensitive to the van der Waals interactions, and it seems possible to match the simulated DL with the diffusion coefficient measured in experiments by varying the depth of the Lennard-Jones potential. Ando and Skolnick [23] argued that the slowdown of diffusion may be due to hydrodynamic interactions, which were not taken into account by [22]. Thus, they showed that accounting for hydrodynamic interactions adequately reproduces the ten-fold reduction of the diffusion coefficient of GFP in the cytoplasm. Another important result of their work is that the details of the macromolecular structure seem to be of negligible importance for BD simulations in a crowded environment, at least for densities up to 350 g l−1. This is shown by comparing the diffusion coefficients in the molecular-shaped system with those in a system where the macromolecules are modeled as spherical particles.
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Living cells are dynamical systems that are maintained by exchanging matter and energy with an environment, and as such are essentially characterized by constant changes of the cell constituents. These changes are due to inflows and outflows of metabolites, their transformation, and production of new proteins and enzymes, which can cause the variation of the packing fraction widely associated with ‘degree of crowding’, and also of the relative concentrations of macromolecules. While it is clear now that the diffusion slows down as a system becomes denser, the effect of macromolecular composition has received almost no attention. The goal of this work is to study the influence of changes in relative concentrations of macromolecules on diffusion, when the volume fraction is kept constant. To accomplish this task, we first perform BD simulations of a simplified two-component system, where the effect of composition can be studied more systematically, and then we compare these results with the results for a model cytoplasm (see figure 1). In both cases we take into account steric, van der Waals, electrostatic, and hydrodynamic interactions, but we model macromolecules as spherical particles for computational efficiency.
2. Model and method Cytoplasm is a rich and diverse environment. For instance, the cytoplasm of E. coli may contain hundreds of different types of macromolecules, and the experimentally available concentrations are often used to investigate the diffusion in crowded environment [21–23]. However, to systematically study the dependence of diffusion on the composition of macromolecules, it is more convenient to first consider a simplified two-component system, and so in this paper we study a system consisting of small and large macromolecules with the hard-core radii σ A = 2.5 nm and σB = 5 nm, which represent tRNA/triphosphate isomerase and RNA polymerase, respectively, and constitute ≈13% and ≈17 % of the volume fraction and c A ≈ 40% and c B ≈ 6% of the number density of E. coli cytoplasm [21, 23]. These two peaks of size distribution are characterized by the ratio of particle concentrations, c A c B = n A n B ≈ 7 (ni is the number of moles). We will use c AB = c A c B = 7 as in some sense representative, and we will compare it with other values of cAB as well as with a realistic cytoplasm model borrowed from [23]. We model macromolecules as spherical LennardJones particles. The Lennard-Jones potential for large spherical molecules [23, 30] was used to describe nonspecific van der Waals interactions. The depth of the potential well was taken as ϵLJ = 0.375 kcal mol−1 in all cases. The electrostatic interactions were calculated using the Debye–Hückel [31] approximation for 2
spheres, with the inverse Debye screening length κ = 0.1 Å−1 and the dielectric constant ε = 78.54. The total charges were q A = −76 and qB = −41 in units of elementary charge, which correspond to tRNA and RNA polymerase, respectively. For such large charges, the Debye–Hückel approximation is not correct, and effective renormalized charges must be used [32]; we used a renormalization factor of 0.2, as in [23]. The hydrodynamic interactions were calculated by the Rotne–Prager–Yamakawa [33, 34] diffusion tensor, with the Ewald summation [35] used to account for the long-range nature of the hydrodynamic forces; the short-range contribution was neglected. It is known that ‘coarse-graining’ of macromolecules tends to overestimate their diffusion coefficients [36]. For this reason, the hydrodynamic radii, which were larger than the hard-core radii, were used to match the bare diffusion coefficients at a given viscosity (ν = 0.01 Poise); in particular, we took R A = 33 nm and R B = 66 nm, which correspond to the bare diffusion coefficients D0, A ≈ 6.58 Å2 ns−1 and D0, B ≈ 3.29 Å2 ns−1, respectively. We recall that a similar approach has been used in other studies [23], showing a satisfactory comparison with atomistic models. BD simulations were performed using a customized BD_BOX code [37, 38]. Particles were packed in a periodic box of size 75 × 75 × 75 nm3. The Ermak–McCammon algorithm [39] with the Iniesta– de la Torre [40] correction was applied to solve the BD equations of motion. The time step was 1 ps or smaller. To gather enough statistics 5 to 10 independent simulations were performed, starting from random initial configurations.
3. Self-diffusion and crossover to anomalous subdiffusion In simulations or experiments, the self-diffusion coefficient can be measured by tracing the particle position (r ) and calculating the mean-square displacement (MSD). Then the Einstein–Helfand relation [41, 42] gives MSD (t ) = 〈 [r (t ) − r (0)]2 〉 = 6Dt α in three dimensions, where α = 1. However, this linear behaviour of MSD at all times occurs only when there is no additional length or time scale in the system, as in the case of the Brownian motion of a single colloidal particle (i.e., in infinite dilution). In a system with two or more characteristic length (or time) scales, the MSD can behave differently, with the exponent α deviating from unity. The anomalous subdiffusion [9, 26, 27] with α < 1 is frequently observed in crowded systems. Let us consider, as an example, the diffusion of macromolecules (or colloids) immersed in an aqueous solution. Apart from momentum relaxation, there is a separation of time scales in MSD (t ) determined by macromolecular collisions. On short time scales, the behaviour of MSD (t ) reflects the viscosity of a solvent,
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Figure 1. Snapshots of BD simulations (a) of a model cytoplasm and (b) of a dense two-component system. The volume fraction occupied by macromolecules is approximately 19% in both cases.
and hence the diffusion coefficient is comparable to the bare diffusion coefficient, D0 = k B T 6πνR , with ν being the viscosity and R being the hydrodynamic radius, as before. At much longer time scales, the macromolecule motion is affected by the collisions with other macromolecules; the diffusion coefficient falls, but the MSD may still behave linearly with time. These two diffusion coefficients are called short- and longtime self-diffusion coefficients, DS and DL, respectively. Another possibility is the anomalous subdiffusion in which molecules are permanently ‘trapped and released’ by their neighbours. In any case, the deviation from the normal short-time diffusion will roughly occur on the time scales determined by macromolecular collisions. The crossover time to anomalous subdiffusion (either transient or permanent) is difficult to calculate analytically, and so we resort to simple estimates. For the ‘free diffusion path’ for a molecule of type k, we write λ k ≈ aρ−1
3
⎛ − ⎜⎜ σk + ⎝
M
⎞
i
⎠
∑x i σi⎟⎟
(1)
where ρ = N V is the total density, x i = Ni N is the concentration, Ni is the number of macromolecules of M type i, N = ∑ Ni is the total number of macroi molecules, and M is the number of different types of macromolecules; σi is the hard-core radius, and a is the proportionality factor, for which we choose a = 1. (Another possible choice is the Wigner–Seitz radius, which gives aWS = (3 4π )1 3. More detailed calculations give a = Γ (4 3) αWS , see [43, 44], which differs by a factor of ≈0.55 from our definition. We choose a = 1 for simplicity, and to get a lower estimate for the time step in point-based BD simulations (see below). Note, however, that this definition may overestimate the maximum time step in molecular-based BD simulations.) In equation (1), the first term is (approximately) the average distance between the molecules, and the second term in the brackets gives the average distance at close contact (assuming that 3
the system is homogeneous). We now use the bare self-diffusion coefficient, D0, i , so for the inter-collision and hence crossover time for a molecule of type i, we get
τi = λ i2 D0, i .
(2)
The crossover time at a constant total packing/volume fraction depends strongly on composition (figure 2). It is larger for smaller molecules, as one might expect, and increases with a decrease in the fraction of bigger macromolecules. The variation of τ with concentration is weaker when the total excluded volume, rather than the packing fraction, is kept constant (lower inset in figure 2). Interestingly, the crossover time is also larger for more polydisperse systems like cytoplasm (upper inset of figure 2). 3.1. Time steps in BD simulations There are two types of BD simulations typically used in the literature. First, there are microscopic or coarsegrained simulations, where the molecular size is taken into account either on the atomistic or the coarsegrained level [21–23]. Second, there are BD simulations where macromolecules are treated as point-like particles [45, 46]. The crossover time sets natural limits for the time steps in these BD simulations (note that there are also other restrictions on the time step, such as intermolecular forces, reaction rates, etc.). Clearly, in microscopic Brownian (and also molecular) dynamic simulations, the time step must be smaller than the crossover time, (i.e., Δt ≪ τ). On the other hand, in point-particle simulations, the time step can be larger than the crossover time (more precisely, larger than the time when the long-time normal diffusion is ‘re-established’). It is important to note that in point-particle BD, the long-time selfdiffusion coefficient must be taken, but it is clearly not possible to take the short-range diffusion coefficient, as there is no way to reproduce the correct long-time behaviour in systems where particles have no size. It may happen that the long-time (t → ∞) behaviour is not reached in certain problems, as for
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Figure 2. Crossover time (τ ) to anomalous subdiffusion for a two-component system and for a cytoplasm model; τ was obtained using equation (2). The hard-core radii σB = 5 nm and σ A = 2.5 nm. The packing fraction was kept fixed, η = (4π 3) ∑i ρi σi3 = 0.186168, where ρi = Ni V is the density of molecules of type i. The upper inset shows the crossover time as a function of the packing fraction for x A = 0.875 and compares it with τ of the same macromolecule in a more realistic cytoplasm model borrowed from [23]. The lower inset shows the crossover time for a fixed excluded volume, Vex(i) = (4π 3) ∑ j ρ j (σi + σ j )3 ≈ 1.5. Notably, the variation of τ is smaller when the excluded volume, rather than the packing fraction, is kept constant. In the lower inset the axis scales and limits are the same as on the main plot.
instance when the crossover length (roughly MSD1 2 (τ)) is comparable to the size of a system. In this case, either microscopic simulations (and the short-time diffusion) can be used, or subdiffusion can be modeled by fractional diffusion [27, 47–50]. However, the latter requires that the exponent, α, of subdiffusion can be obtained, which may not generally be the case [51]. In our BD simulations, we took Δt = 1 ps, and in some cases even smaller values of Δt were used, which is well below the calculated crossover times (see figure 2).
4. Results of BD simulations 4.1. MSD and crossover time: dependence on packing fraction Despite its simplicity, the estimated crossover time, equation (2), agrees surprisingly well with the results of BD simulations. Figure 3 shows the MSD versus time for a two-component system with different packing fractions, where symbols denote the crossover time, τ , calculated by equation (2). τ decreases when the volume fraction increases, and coincides approximately with the time when the MSD starts to deviate from the normal diffusion (i.e., where MSD ∼ t α , α = 1, for t < τ ). Figure 3 also demonstrates that diffusion slows down when the volume fraction increases. This is in accordance with the experiments [28, 29] and other simulations [22, 23]. Note that both the diffusion coefficient and the crossover time decrease. It is not clear at this stage whether the anomalous subdiffusion (α < 1) will persist or the normal diffusion with a smaller diffusion coefficient will re-establish at long 4
Figure 3. MSD as a function of time for a two-component dense system from BD simulations. The MSD is shown for a few values of the volume fraction (100 × η where η = (4π 3) ∑i ρi σi3 with ρi being the density and σi being the radius of a molecule of type i = {A, B}). The hard-core radii are σ A = 2.5 and σB = 5 nm, and the concentration x A = NA N = 0.785. The symbols show the crossover time to anomalous subdiffusion calculated from equation (2). The thin dashed line shows the linear behaviour, MSD ∼ t , corresponding to normal diffusion.
times. Our preliminary results suggest, however, that the latter is more likely the case, with the window of anomalous subdiffusion extending from t = τ to about a few to tens of microseconds. For instance, for the volume fraction 19% (a less computationally demanding case with a relatively small window of transient subdiffusivity), we observe the long-time normal diffusion for t > τL ≈ 4.7 μs as follows from the saturation of MSD/t. For the σ A = 2.5 nm macromolecule, the long-time self-diffusion coefficient is D L, A ≈ 1.04 Å2 ns−1, which is more than sixfold smaller than the diffusion coefficient in infinite dilution (D0, A ≈ 6.58 Å2 ns−1). 4.2. Influence of composition To study the effect of composition, one can keep the number density, ρ = N V , fixed and vary the concentration of different macromolecules, x i = Ni N , where i = {A, B} for a two-component system. However, the density alone does not distinguish between molecular sizes, and thus does not tell us how crowded the system actually is. Therefore, it seems more natural to keep the packing fraction constant, η = (4πρ 3) ∑i x i σi2 = const, when varying the concentrations, xi. To do this consistently, it is convenient to rewrite, for a two-component system, η = (v A V )(NA + r 3NB ), where r = σB σ A and v A = (4π 3) σ A3 is the volume of molecule A. To vary xi at fixed η amounts to varying NB and changing NA according to NA (NB ) = N A(max) − r 3NB , where N A(max) = Vη v A . Clearly, NB can be varied from zero to NB(max) = N A(max) r 3, where NB(max) implies NA = 0. In what follows, we took N A(max) = 1200, which with corresponds to η = 0.186168 ≈ 0.19 V = (75 nm)3 and σ A = 2.5 nm.
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Figure 4. MSD as a function of time for a two-component dense system from BD simulations. The MSD is shown for a few values of concentration of smaller macromolecules (x A ). The volume fraction 100 × η ≈ 19% , where η = (4π 3) ∑i ρi σi3 ≈ 0.19, ρi is the density and σi the radius of a molecule of type i = {A, B}. The macromolecule radii are σ A = 2.5 and σB = 5 nm. The symbols show the crossover time to anomalous subdiffusion calculated from equation (2). The thin dashed line shows the linear behaviour, MSD ∼ t , corresponding to normal diffusion.
Figure 4 shows the influence of composition for a few values of the concentration of smaller macromolecules, x A = NA N . The diffusion clearly slows down when xA increases. This correlates with a decrease of the crossover time with increasing xA (figure 2). By keeping the packing fraction constant and increasing the number of A (smaller) molecules, the average distance between the molecules decreases (see equation (1) and figure 2), and the excluded volume increases. The volume excluded by i molecule to a j molecule is vex(ij) = (4π 3)(σi + σ j )3. For instance, consider an A molecule as a test particle. The volume excluded to it by another A molecule is vexAA = 8v A ; the volume excluded to it by a big (B) molecule is vexAB = 27v A (vA is the volume of the A molecule). Trivial arithmetic shows that replacing one B molecule by 8 A molecules keeps the volume taken up by all molecules constant, but increases the excluded volume by 64 27 ≈ 2.4. It is likely that this increase in the excluded volume and a decrease in the free diffusion length are responsible for the diffusion slow-down when the number of smaller molecules increases in a crowded system. The allusion to this is that the crossover time varies less with the concentration when the excluded volume, rather than the packing fraction, is kept constant (the lower inset in figure 2). Physically, adding a small number of bigger (B) macromolecules mainly creates a few steric obstacles for other molecules, while some distance away from them, the system remains relatively dilute. On the contrary, adding many smaller macromolecules (of the same total volume) makes the system effectively denser and leads to the decrease in diffusion. Surprisingly, the slow-down in diffusion caused by a change in concentration (at the same volume fraction) is comparable or even stronger than the slow5
Figure 5. Comparison of the MSD of a diffusing macromolecule of size 2.5 nm in a model cytoplasm and in a twocomponent system. The diffusion is faster in the cytoplasm although the packing fractions of both systems are comparable. The cytoplasm model is taken from [23] and has a volume fraction of 19.2%. The two-component system consists of only two types of macromolecules and has volume fractions of 18.6% and 11.6%, and the concentration of smaller molecules x A = 0.875.
down induced by an increase of the packing fraction (compare figures 3 and 4). This turns out to be true also for more complex systems. Figure 5 shows that the (middle-time) diffusion in the cytoplasm is faster than in a two-component system at comparable packing fractions. Similarly, the diffusion is comparable in the cytoplasm and in the two-component system, although the cytoplasm is over 50% more crowded than the two-component system.
5. Conclusions We have studied the diffusion of macromolecules in a crowded environment. Using BD simulations, we characterized diffusion by calculating the MSD. At short times, the MSD depends linearly on time, MSD ∼ t α , where α = 1, and we observe normal diffusion. At longer times, the collisions between macromolecules become important, the MSD starts to deviate from the normal behaviour, and the diffusion develops into being anomalously slow (a so-called subdiffusion, α < 1). We have estimated the crossover time to anomalous subdiffusion from the free diffusion length (equations (1) and (2)). In defiance of its simplicity, this estimate agrees surprisingly well with the results of BD simulations (figure 3). In accordance with other theoretical [22, 23] and experimental [28, 29] studies, the diffusion slows down when the volume fraction increases (figure 3). Surprisingly, however, the diffusion and crossover time seem to depend strongly on the molecular composition as well. The diffusion slows down when the number of smaller molecules increases while the volume fraction is kept constant (figure 4). This is because the average distance between the macromolecules becomes shorter, effectively rendering the
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system denser. For the same reason, the crossover time to anomalous diffusion becomes smaller (figure 2). The slow-down of diffusion induced by a change in composition is comparable to or can even be stronger than the slow-down caused by an increase in the volume fraction. A striking example is the cytoplasm model, in which the (middle-time) diffusion turns out to be faster than in a two-component system at comparable volume fractions (figure 5). This study suggests that the volume fraction alone is not a unique measure of how crowded a system is with respect to diffusion. Indeed, diffusion depends sensitively on molecular composition, and can in fact be faster in systems with higher volume fractions. Weiss et al have proposed using an exponent of the anomalous subdiffusion (i.e., α in MSD ∼ t α ) as a measure of crowdedness [9]. A crossover time from the short-time normal diffusion to the anomalous subdiffusion, τ, may serve as an additional parameter characterizing the degree of crowding. It is interesting to note, however, that both τ and α depend on molecular sizes and on the interaction between molecules (the latter is not taken into account in a simple estimate (2)). Thus, the so-defined ‘degree of crowding’ can be different for two systems with the same volume fraction but with different intermolecular interactions. More importantly, it will differ for different molecules in the same system. For instance, as figure 2 suggests, a system can look crowded in the eyes of a large molecule, but can still be relatively dilute for smaller molecules.
Acknowledgments This work was supported by BMBF Grant No. 031A129. The authors acknowledge the computing time granted on supercomputers JUROPA and JUDGE of Jülich Supercomputing Centre (JSC). SK is grateful to Paweł Zieliński and Maciej Długosz for fruitful discussions and support with BD_BOX, and to Pradeep Burla for technical assistance.
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