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Crowding Effects on Protein Association: Effect of Interactions between Crowding Agents Jun Soo Kim and Arun Yethiraj* Department of Chemistry and Theoretical Chemistry Institute, UniVersity of Wisconsin, Madison, Wisconsin 53706, United States ReceiVed: July 29, 2010; ReVised Manuscript ReceiVed: NoVember 20, 2010
The cell cytoplasm is a dense environment where the presence of inert cosolutes can significantly alter the rates of protein folding and protein association reactions. Most theoretical studies focus on hard sphere crowding agents and quantify the effect of excluded volume on reaction rates. In this work the effect of interactions between the crowding agents on the thermodynamics of protein association is studied using computer simulation. Three cases are considered, where the crowding agents are (i) hard spheres, (ii) hard spheres with additional attractive or repulsive interactions, and (iii) chains of hard spheres. Reactants and products of the protein association are modeled as hard spheres. Although crowding effects are sensitive to the shape of the reaction product, in most cases the excess free energy difference between the product and reactants (nonideality factor) is insensitive to the interactions between crowding agents, due to a cancellation of effects. The simulations therefore suggest that the hard sphere model of crowding agents has a surprisingly large regime of validity and should be sufficient for a qualitative understanding of the thermodynamics of crowding effects when the interactions of associating proteins with crowding agents other than excluded volume interactions are not significant. 1. Introduction The crowded nature of the cytoplasm can play an important role in the rates of biochemical reactions. Examples include protein association reactions, nonspecific binding to DNA, and the release of repressors specifically bound to DNA. The volume occupied by crowding agents is significant, and there has been considerable effort devoted to understanding the impact of excluded volume crowding interactions on the diffusion, binding, and reaction of globular proteins.1-8 Most theoretical work focuses on hard sphere models for the crowding agents, and in this work we investigate the effect of interactions between crowding agents. The most well studied model is the association of hard sphere proteins in the presence of hard sphere crowding agents. Smoluchowski theory and dynamic simulations of this system9,10 show that the overall reaction rate (kF) can be accurately represented as the sum of two contributions as proposed by Minton,4,5 i.e.
1 1 1 ) + kF kD kts
(1)
where kD and kts are, respectively, the rate constants for an encounter and a (transition-state-limited) reaction between reactants. The presence of crowding agents reduces the diffusion constant of the reactants but increases the equilibrium constant for the association: The overall reaction rate reflects a balance between the decreased probability of an encounter between reacting species and an increased thermodynamic driving force for association. Minton4 used scaled particle theory (SPT)11 for hard spheres to calculate the effect of crowding on these two contributions. Experiments have shown that crowding effects can either decrease or increase the reaction rate. For example, protein-protein association rates are decreased in the presence of high concen-
trations of crowding agents when compared to that in buffer solutions.12,13 In the presence of polymeric crowding agents, however, the behavior depends strongly on whether the solution is dilute, semidilute, or concentrated.13 On the other hand, the self-association of human spectrin is enhanced by high concentrations of polyethylene glycol (PEG),14 and the formation of amyloids by human apolipoprotein C-II is accelerated by macromolecular crowding.15 Recently, Phillip et al.16 studied the heterodimerization equilibrium for the binding of TEM1 β-lactamase with β-lactamase inhibitor protein and barnase with barstar. Interestingly, they found that crowding (with PEG and dextran) had only a small effect on protein-protein heterodimerization reactions. Although experiments can be qualitatively understood on the basis of existing theoretical ideas,17 a quantitative understanding is lacking.18 Theoretical treatments often use the hard sphere model for crowding agents and fit the parameters to experiment. Agreement can be obtained with a suitable hard sphere diameter, although this is not entirely satisfactory. For example, as pointed out by Elcock,18 three different hard sphere diameters were obtained for the same dextran crowding agent when theory was fit to three different experiments.19-21 A desire to go beyond simple models has motivated ambitious simulations of the entire cytoplasm using computer simulations.22,23 There are two classes of theoretical approximations that merit testing: The first is the model for the crowding agents, and the second is the approximate theory to describe crowding effects (for a given model). The majority of theoretical work has focused on hard sphere crowding agents mimicking the excluded volume effect, and it is important to determine, as has been previously emphasized,24 if this is representative of real systems. In particular, what is the effect of the local structure of the crowding medium on crowding effects? This point can be addressed by investigating crowding agents with attractive and repulsive interactions in addition to the hard sphere potential. The interactions other than the excluded volume interactions
10.1021/jp107123y 2011 American Chemical Society Published on Web 12/17/2010
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have been included to study the crowding effects on thermodynamic stability in previous work,23,25 and the work by Elcock25 is discussed together with our results. The second point is that the SPT (and other theories) are approximate, even for hard spheres. It is important to test the accuracy of the approach by comparison to exact (for the hard sphere model) simulation results. Interestingly, the approximations inherent to the SPT have never been tested via comparison with exact computer simulations for model systems. The effect of crowding on the two contributions to the overall reaction rate can be investigated independently. The encounter reaction rate depends only on the diffusion coefficient of the reactions, i.e, kD ) 4πDσ, where D is the diffusion constant and σ is the hard sphere diameter of the reactants. The transitionstate-limited rate can be written as4,5
kts )
kts0Γ
(2)
where kts0 is the rate constant in the absence of crowding agents and Γ, called the nonideality factor, is a purely thermodynamic quantity. In this work we only focus on the crowding effect on the thermodynamics of protein association: the nonideality factor, Γ, and thus kts. We have two broad objectives. The first is to test the approximations inherent to the SPT for hard sphere crowding agents via exact computer simulations. The second is to investigate the effect of the nature of the crowding agents and the geometry of the product on the reaction rates. We consider several cases, where the crowding agents are (i) hard spheres, (ii) hard spheres with additional attractive or repulsive interactions between them, and (iii) chains of hard spheres. We calculate the nonideality factor exactly, using Widom’s test particle method26 in a Monte Carlo simulation. For hard sphere crowding agents the SPT is in good agreement with simulation results for the equilibrium constant for the association of certain spherical and spheroidal products. On the other hand, the theory is not applicable to the nonconvex products such as a tangent dimer, whose nonideal contribution to the equilibrium constant is quite different from that of a spheroid. We suggest alternative theories for the tangent dimer that have an analytic closed form, no adjustable parameters, and are in quantitative agreement with the simulation results. We find that when the reaction product is a sphere or a spheroid, the nature of the crowding agents has little impact on the nonideality factor. When the reaction product is a tangent dimer, the nonideality factor is sensitive to interactions between crowding agents. Overall, the hard sphere model provides a reasonable description of the nonideality effects for all the cases considered. The rest of this paper is organized as follows. The model and simulation method are described in section 2, the results are presented in section 3, and some conclusions are presented in section 4. 2. Model and Simulation Methods We consider a test system where the reactants are hard spheres of diameter σ and the interaction between the reactants and crowding agents is a hard sphere interaction with a hard sphere diameter of σ. We consider three types of products (see Figure 1). In model 2 the product is a tangent dimer; in models 1 and 3 the product is a sphere and a spheroid, respectively, with the sphere diameter and spheroid length chosen so that the volume of each is the same as that of the tangent dimer. For a spheroid
Figure 1. Left: Pictorial representation of the model showing the reactants (shaded spheres), product (shaded dimer), and crowding agents (open spheres). Right: The three models for the product. In model 2, the reactants associate to form a dimer. Models 1 and 3 are, respectively, an effective sphere and an effective spheroid with the same volume as that of the dimer.
where the diameter of the hemispherical caps is dσ and the cylinder length is l
d3
[ 21 + 43 dσl ] ) 1
(3)
where the diameter of the spherical product can be obtained by setting l ) 0. Proteins can certainly associate to form asymmetrical products. Our model is most suited to dimerization accompanied by a negligible conformational change where the product is symmetric and not very elongated. We consider two classes of crowding agents: spherical and polymeric. For the spherical crowding agents the interaction between crowding agents is given by
βu(r) ) U0
e-κ(r/σ-1) r/σ
(4)
for r > σ and βu(r) ) ∞ for r < σ. The parameters U0 and κ determine the strength and range of the interactions, respectively, and hard sphere crowding agents are recovered for U0 ) 0. The polymeric crowding agents are freely jointed hard sphere chains with n monomers (n ) 20 in this work). In all cases the volume fraction, φ, is defined in terms of the hard sphere diameter, i.e., φ ≡ πFσ3/6, where F is the number density of spheres or monomers (in the case of polymeric crowding agents). Our model is aimed to mimic typical conditions in cells where the crowding agents are primarily globular proteins. Within the Debye-Hu¨ckel model, the inverse Debye screening length varies from 5 to 30 Å for salt concentrations ranging from 10 to 250 mM. If the protein diameter is 4 nm, then κ ≈ 1.33-8, and we investigate two cases of κ ) 1 and 3 (κ ) 8 results in a very short ranged potential, and the results are not expected to be very different from those of hard spheres). Similarly, the strength of the interaction is approximately q2lB/σ, where lB ()7.14 Å for water at room temperature) is the Bjerrum length and q is the charge on a protein. In the range q ≈ 1-10, this gives U0 ≈ 0.1-10, and we investigate two cases of U0 ) 1 and 5. For attractive interactions between crowding agents, we wish to avoid the two-phase region27 and use U0 ) -0.8 and -1 for κ ) 3 and 1, respectively. The structure of the crowding media, i.e., pair correlation function between crowding agents, is sensitive to the pair interactions. Figure 2 depicts the radial distribution function between crowding agents for κ ) 3, φ ) 0.3, and various values of U0. There are large differences between g(r) in the various cases, especially near contact (r ) σ), which is the important
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Figure 2. Radial distribution function between crowding agents for κ ) 3 and φ ) 0.3.
range of r (because the interaction between the reactants or products with the crowding agents is purely hard core). Minton has estimated the effect of crowding on the transitionstate-limited rate constant, kts.4,28 For bimolecular reactions where the transition-state complex excludes the same volume to the crowding agents as the fully formed complex28
γ12 Γ) γ2
(5)
where γ1 is the activity coefficient of a reactant hard sphere and γ2 is the activity coefficient of the product. We obtain the activity coefficients (and thus Γ) from Monte Carlo simulations in the canonical ensemble (where the temperature, volume, and number of molecules of the crowding agents are fixed). The reactants and products are at infinite dilution in this system. The excess (over ideal gas) chemical 26 potential, µex i , of species i is given by
µiex ) -kBT ln〈e-βψi〉
(6)
where kB is Boltzmann’s constant, T is the temperature, β ) 1/kBT, and ψi is the potential of interaction between a test particle of species i and the crowding agents. The average is over all configurations of the crowding agents. For hard particles the average 〈e-βψi〉 has an intuitive interpretation as an insertion probability: since ψi is either infinite or zero, e-βψi ) 0 if the test particle overlaps with a sphere of the fluid and e-βψi ) 1 otherwise. The average value is therefore the probability that a test particle is inserted successfully into the fluid, and we calculate this from simulations for the reactants and products. If pi is the insertion probability of species i, then
µiex
) -kBT ln pi
p2 p12
several (105) attempts are made to insert a particle (both reactant and product) at randomly chosen positions. For the product species 100 randomly chosen orientations are attempted at each position. Insertion probabilities are calculated over 10-100 runs, each of which consists of 106 attempted moves. Averages are obtained from a standard deviation about the mean of these 10-100 runs. The reactant is a sphere, and we consider three types of product species: a tangent dimer, a sphere, and a spheroid (cylinder capped with a hemisphere at each end). Once the insertion probabilities are obtained, Γ is determined from eq 8. The simulations are a standard application of the Metropolis algorithm. For spherical crowding agents, initial configurations are generated by randomly inserting spheres into a box with periodic boundary conditions in all directions. The system is then evolved via random translation of the spheres with the maximum displacement chosen so that about 50% of the attempted moves are accepted. For polymeric crowding agents, initial configurations are generated by the growth and equilibration algorithm.31 The system is evolved using a combination of Monte Carlo moves; the procedure is identical to that described elsewhere.32,33 3. Results and Discussion 3.1. Hard Sphere Crowding Agents. The SPT provides approximate expressions for p1 and p2. For hard spheres
-ln p1 ) -ln(1 - φ) + 7
φ 15 φ + 1-φ 2 1-φ
(
(8)
Note that a similar method has been employed in previous work to calculate the crowding effect on the thermodynamic stability of protein folding and protein association.23,29,30 The insertion probabilities for reactants and products are obtained as follows. A system of crowding agents is simulated using a standard Metropolis scheme. For every 1000 attempted moves,
)
2
+ 3
( 1 -φ φ )
3
(7)
By comparing this equation to the definition of the activity coefficient, i.e., µex i ) kBT ln γi, we obtain γi ) 1/pi and therefore
Γ)
Figure 3. Comparison of the SPT prediction (dashed lines) to the simulation results (symbols) for the nonideality factor. The solid line corresponds to Γ ) g(σ+), and the dotted line is the truncated virial expansion for model 2 (eq 14).
(9)
and for a spheroid where the diameter of the hemispherical caps is dσ and the cylinder length is l4
[ (
-ln p2 ) -ln(1 - φ) + 3d 1 +
[ (
l* 9d2 1+ 2 2
)
2
l* φ + + 3d2(1 + l*) + 2 2 1-φ φ 2 φ 3 +6 +6 (10) 1-φ 1-φ
)
](
]
)
(
)
where l* ) l/dσ and d is obtained from eq 3. The SPT is accurate for the nonideality factor for convex body products as long as they are not very nonspherical. Figure 3 compares the predictions of the SPT to simulation results for the three products, i.e., tangent dimer, sphere, and spheroid, with
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Figure 4. Comparison of the SPT prediction (lines) to the simulation results (symbols) for the nonideality factor for spheroid products. From top to bottom, the curves (and symbols) correspond to l* ) 0, 2/3, 1, 3/2, and 2, respectively.
d ) 1. The SPT is quite accurate for both the spheroid and sphere models except perhaps at the highest volume fraction tested. There are significant differences in the value of Γ between these models and the tangent dimer model, more than 100% for the highest volume fraction. For the particular case of a hard sphere dimer in a hard sphere fluid (see the Appendix)
Γ ) g(σ+)
(11)
where g(σ+) is the value of the radial distribution function at contact. Physically, the probability of two reactants being in contact (relative to being far away) is equal to the probability of two spheres at contact relative to an ideal gas, which is exactly equal to the contact value of the radial distribution function. The prediction of the Carnhan-Starling result for g(σ+) where
g(σ+) )
1 - φ/2 (1 - φ)3
(12)
is shown as the solid line in Figure 3. This approximation is in excellent agreement with the simulation results for the tangent dimer over the entire range of volume fractions. Also shown in the figure are the predictions of a virial expansion for the tangent dimer model.4 If only the first term of the virial expansion is retained
ln γi ≈ BiF
(13)
where Bi is the volume excluded by species i to the crowding agents. For reactants and the dimer, Bi can be calculated exactly: B1 ) 4πσ3/3 and B2 ) 9πσ3/4. Since ln Γ ) 2 ln γ1 - ln γ2 ) (2B1 - B2)F ) 5πσ3F/12, we have
5 ln Γ ) φ 2
(14)
This approximation is in excellent agreement with the simulations for φ e 0.25. The accuracy of the SPT diminishes as the product becomes more nonspherical. Figure 4 compares the SPT predictions4 to simulation results for the nonideality factor of spheroidal products for various values of l* (with d given by eq 3). For l* ) 0 and 2/3 the theory is in excellent agreement with the
simulations. As l* is increased, however, the theory begins to underestimate the value of the nonideality factor, although it captures the qualitative trend that Γ decreases with increasing l*. The SPT is therefore more accurate for products with low asphericities than for long skinny products. The conformational change of proteins is usually very limited upon association, and it is reasonable to consider that the product is close to either a tangent dimer or a spheroid with l* ) 2/3, rather than a sphere or more aspherical spheroids. Therefore, the deviation of the SPT for significantly aspherical products is not a significant drawback of the theory. 3.2. Effect of Attractions and Repulsions between Crowding Agents. The nonideality factor is relatively insensitive to the interactions between the crowding agents. This is a surprising result because the pair correlation function is a strong function of these interactions. Figure 5 depicts ln Γ as a function of φ for the six different values of U0 and κ investigated and for models 1-3. The different symbols are hard to discern, but the main point is that for models 1 and 3 the interactions between crowding agents have a negligible impact on the nonideality factor. In all cases, introducing attractions increases Γ and introducing repulsions decreases Γ, but except for model 2, these differences are quite small. (Results are not reported for φ > 0.3 for U0 ) 5 because the system crystallizes.) Interestingly, there is a strong effect of interactions between crowding agents on the insertion probabilities p1 and p2, but these effects cancel out when we consider the ratio Γ ) p2/p12. Figure 6 depicts ln p1 and ln p2 as a function of φ for κ ) 3. Comparing the filled symbols or open symbols shows that the structure of the crowding media has a significant effect on the insertion probabilities of the reactant or the product. The results for ln p2 shown are for model 3, although similar behavior is seen for the other models. The effect of these interactions cancels out, however, when one considers the difference ln Γ ) ln p2 - 2 ln p1 (see Figure 5). In previous work,25 Elcock studied the crowding effect on the escape of a protein from the GroEL cage. The free energy changes of the escape due to crowding were compared for steric and interacting crowding agents. Consistent with our simulation results, the free energy changes do not depend significantly on the different types of interactions between crowding agents. In this work we only consider the reactants and products interacting with crowding agents via the exclusion volume interactions. McGuffee and Elcock performed the computer simulations of the entire cytoplasm.23 In their simulations all proteins interact with each other in the same way, by either only the excluded volume interactions or the electrostatic interactions in addition to the excluded volume interactions, whether it is a reactant or crowding agent. For many of the proteins shown in their data, the association free energy calculated at the given protein concentration is altered significantly depending on the interactions between particles. This suggests that the types of interactions between reactants and crowding agents may change the crowding effect on the thermodynamic stability of protein association. 3.3. Polymeric Crowding Agents. The nonideality factor with hard-chain polymeric crowding agents is very similar to that of hard spheres. Figure 7 depicts ln Γ as a function of the volume fraction of crowding agents for the three models of products. Also shown are simulation results for hard sphere crowding agents. In all cases the differences between the nonideality factor with polymeric and hard sphere crowding agents are very small. As was the case with attractive and repulsive spherical crowding agents, the individual insertion
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Figure 5. Effect of attractive and repulsive interactions between crowding agents on the nonideality factor for (a) model 1, (b) model 2, and (c) model 3. The symbols denote cases with U0 ) -0.8 and κ ) 1 (O), U0 ) -1 and κ ) 3 (0), U0 ) 1 and κ ) 1 (b), U0 ) 1 and κ ) 3 (9), U0 ) 5 and κ ) 1 (2), and U0 ) 5 and κ ) 3 (1). The results for model 3 are for l* ) 2/3. The solid lines are simulation results for hard sphere crowding agents (from Figure 3).
Figure 6. Effect of interactions on the logarithm of the insertion probabilities for κ ) 3 and U0 ) -0.8 and 5. The ln p2 shown is for model 3.
Figure 7. Simulation results for the nonideality factor with polymeric crowding agents for models 1 (b), 2 (9), and 3 (2). Lines are simulation results for hard sphere crowders at the same volume fraction (numbers denote the model type).
probabilities p1 and p2 are different with polymeric as opposed to hard sphere crowding agents, but the nonideality factor is not. 4. Summary and Conclusions We present exact computer simulation results for the thermodynamics of crowding effects on model protein association reactions. Our primary objective is to investigate the impact of the interactions between crowding agents on the nonideality
factor, Γ, defined as p2/p12, where p2 is the insertion probability of the product and p1 is the insertion probability of a reactant. We consider spherical crowding agents with a hard sphere plus either an attractive or a repulsive “tail” interaction and polymeric crowding agents composed of chains of hard spheres. In all cases the reactants and products interact with the crowding agents via only excluded volume interactions. Our main conclusion is that the nonideality factor is insensitive to the interactions between the crowding agents even though the radial distribution function is very different for the various cases studied. In fact, the insertion probabilities p1 and p2 do depend on the interactions between crowding agents, but the ratio Γ ) p2/p12 does not, probably due to a cancellation of effects. In other words, the probability of inserting two reactants into the solution depends on the interactions between crowding agents, but the change in free energy for converting these into a product does not. We use the simulations to test the predictions of the scaled particle theory of Minton4 for crowding effects in hard sphere solutions. The SPT provides a quantitatively accurate framework for studying crowding effects in proteins, as long as the products are not too skinny. Therefore, any disagreement between theory and experiment must be attributed to the different geometry of the formed product, rather than a breakdown of the theory itself. We therefore conclude that the hard sphere model should provide a reasonable description of the effect of crowding on the thermodynamics of protein association reactions, when the interactions of the associating proteins with crowding agents other than excluded volume interactions are not significant. As a caveat there are a number of factors that we do not consider that could be important in real systems. For example, the diffusion coefficient can be a very strong function of the interactions between crowding agents. Using an Enskog approach, we estimate that the diffusion coefficient can be different by almost an order of magnitude for the different models considered in this work (depending on the strength of the interactions and the volume fraction). We also do not consider the effect of size and interaction polydispersity, nonspherical crowding agents, crowding agents with patchy anisotropic
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interactions, or attractive interactions between the associating proteins and the crowding agents. Among them, the effect of attractive interactions between the association proteins and crowding agents could be particularly interesting. In recent work McGuffee and Elcock23 showed that the dimerization free energy is altered significantly for some of the protein dimers when the electrostatic interactions in addition to the excluded volume interactions are included between all proteins. These extensions are useful and represent future directions of this work. Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant CHE0717569. We thank Dr. A. P. Minton for useful discussions and for sharing his MATLAB code for calculation of the activity coefficient of hard convex particles. Appendix Here we derive the result Γ ) g(σ+) for hard sphere reactants in a system of identical hard sphere crowding agents. For a system with N + 2 identical particles, the radial distribution function is given by
g(rN+1, rN+2) ) (N + 2)(N + 1) F2
∫ dr1...drN exp(-βUN+2) ZN+2
∫ dr1...drN+2 exp(-βUN+2)
(15)
(16)
We define the quantity ψ1(J + 1) as the potential of interaction between particle J + 1 and particles 1-J. Similarly, ψ2(J + 1, J + 2) is the potential of interaction between two particles, J + 1 and J + 2, and particles 1-J. In terms of these quantities, we have
UN+2 ) UN+1 + ψ1(N + 2) UN+1 ) UN + ψ1(N + 1) UN+2 ) UN + ψ2(N + 1, N + 2)
(17)
Using Widom’s method,26 we can write ZN+2 in terms of ZN as follows:
∫ dr1...drN+2 exp(-βUN+2) ) ∫ dr1...drN+2 exp(-βUN+1) exp(-βψ1(N + 2))
ZN+2 )
)VZN+1〈e-βψ1(N+2)〉N+1
(18) where the average is over the N + 1 particle system. Using the same result for ZN+1
ZN+2 ) V2ZN〈e-βψ1(N+2)〉N+1〈e-βψ1(N+1)〉N
∫ dr ...dr 1
N
(N + 2)(N + 1) × F2
exp(-βUN) exp(-βψ2(N + 1, N + 2)) ZN+2
)
〈e-βψ2(N+1,N+2)〉N (N + 2)(N + 1) N2 〈e-βψ1(N+2)〉N+1〈e-βψ1(N+1)〉N )
(20)
〈e-βψ2(N+1,N+2)〉N 〈e-βψ1(N+2)〉N+1〈e-βψ1(N+1)〉N
where we have used eq 19 in the first step and the last step follows from the thermodynamic limit, [(N + 2)(N + 1)]/N2 ) 1 for N f ∞. In the thermodynamic limit, both the averages in the denominator of eq 20 are equal to the (ensemble average) probability of inserting a sphere into the hard sphere fluid, which we have defined as p1, i.e.
〈e-βψ1(N+2)〉N+1 ) 〈e-βψ1(N+1)〉N ) p1
(21)
If we choose rN+1 and rN+2 so that |rN+1 - rN+2| ) σ, then
where ri is the position of the ith sphere, UN+2 is the potential of interaction of a system with N + 2 spheres, and ZN+2 is the canonical ensemble partition function, i.e.
ZN+2 )
g(rN+1, rN+2) )
(19)
Substituting the expression for UN+2 in terms of UN into eq 15, we have
〈e-βψ2(N+1,N+2)〉N ) p2
(22)
is the (ensemble average) probability of inserting a dimer into the hard sphere fluid. With these substitutions, eq 20 simplifies to
g(σ+) )
p2 p12
)Γ
(23)
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JP107123Y
