# Theory and simulations for crowding-induced changes in stability of proteins
with applications to $\lambda$ repressor

Natalia A. Denesyuk  Biophysics Program, Institute for Physical Science and
Technology, University of Maryland, College Park, MD 20742  D. Thirumalai
Department of Chemistry
University of Texas at Austin, Austin, TX 78731

###### Abstract

Experiments and theories have shown that when steric interactions between
crowding particles and proteins are dominant, which give rise to Asakura-
Oosawa depletion forces, then the stabilities of the proteins increase
compared to the infinite dilution case. We show using theoretical arguments
that the crowder volume fraction ($\Phi_{C}$) dependent increase in the
melting temperature of globular proteins, $\Delta
T_{m}(\Phi_{C})\approx\Phi_{C}^{\alpha}$ where
$\alpha=\frac{1}{(3\nu_{eff}-1)}$. The effective Flory exponent, $\nu_{eff}$,
relates the radius of gyration in the unfolded state to the number of amino
acid residues in the protein. We derive the bound 1.25 $\leq\alpha\leq$ 2.0.
The theoretical predictions are confirmed using molecular simulations of
$\lambda$ repressor in the presence of spherical crowding particles. Analyses
of previous simulations and experiments confirm the predicted theoretical
bound for $\alpha$. We show that the non-specific attractions between crowding
particles and amino acid residues have to be substantial to fully negate the
enhanced protein stabilities due to intra protein attractive Asakura-Oosawa
(AO) depletion potential. Using the findings, we provide an alternate
explanation for the very modest (often less than 0.5 Kcal/mol) destabilization
in certain proteins in the cellular milieu. Cellular environment is
polydisperse containing large and small crowding agents. AO arguments suggest
that proteins would be localized between large (sizes exceeding that of the
proteins) crowders, which are predicted to have negligible effect on
stability. In vitro experiments containing mixtures of crowding particles
could validate or invalidate the predictions.

## Introduction

The crowded cellular environment could profoundly influence many aspects of
interest in biophysics, such as, the folding of proteins Minton05BJ ;
Elcock10PlosCompBiol , RNA Pincus08JACS ; Kilburn10JACS ; Denesyuk11JACS ;
Cheung13COSB ; Jeon16SoftMatter , the formation of oligomers OBrien11JPCL ,
binding of intrinsically disordered proteins Zosel20PNAS . This realization
has lead to great efforts to understand crowding-induced folding of proteins
Minton05BJ ; Cheung05PNAS ; Dhar10PNAS , and subsequently RNA Pincus08JACS ;
Kilburn10JACS ; Denesyuk11JACS ; Denesyuk13BioPhysRev ; Tyrrell13Biochem ;
Tyrrell15Biochem ; Strulson14RNA ; Leamy16QRB . In most of the in vitro
studies the effects of macromolecular crowding is mimicked by monodisperse
particles, which are not only well controlled model systems but also can be
considered as an idealization of the cellular environment. These studies have
shown that the crowding agents enhance the stability, relative to what is
found under infinite dilution conditions. The explanation for the increased
stability can be traced to a remarkable paper in 1954 by Asakura and Oosawa
(AO) Asakura54JCP  and subsequently in a study published four years later
Asakura58JPolySci . They showed that the crowding particles (modeled as hard
objects) induces an effective short-range attraction between (depletion
potential). This concept when applied to folding of globular proteins suggests
that (predominantly) the entropy of the unfolded state should decrease.
Consequently, the crowding should enhance the stability of the folded protein.

In contrast to the ESM, in several experimental studies it has been argued
that weak soft-interactions (also referred to as chemical interactions),
presumably between crowding particles and the residues in the polypeptide
chains negate the stabilizing effects due to ever present hard core
interactions Drishti19ChemRev ; Sarkar13PNAS ; Sapir15CurrOpinColl ;
Sapir14JPCL ; Danielsson15PNAS . Many of these studies imply that this
behavior is universal in the sense destabilization due to crowding is the
norm, especially under in vivo conditions. Thus, there is a need to
theoretically describe folding in a crowded environment, which is complicated
because the interplay of a number of factors such as variations in the sizes,
shapes, volume fraction ($\Phi_{C}$) of the crowding agents, and non-specific
attractive interactions between the crowding particles and the proteins
determines the fate of proteins. As a result crowding could increase,
decrease, or leave unaltered the stabilities of proteins relative to their
values when $\Phi_{C}=0$ Cheung05PNAS ; Miklos11JACS ; Ghosh10BJ ;
Ebbinghaus11JPCL ; Kang15PRL .

Regardless of the complexity of the interactions governing the fate of
polypeptide chains in a crowded environment, to a first approximation, the
most important effect is the exclusion of the volume occupied by the
macromolecules to the protein of interest. If excluded volume interactions
between the crowding particles and the proteins dominate then the stability of
proteins is enhanced compared to $\Phi_{C}=0$ Minton05BJ ; Cheung05PNAS . As
noted earlier this is die to the ESM, which takes into account the greater
crowding-induced suppression of the conformational fluctuations of the
unfolded state relative to those in the native folded state, as Minton showed
in an important study Minton05BJ .

Inspired in part by an experimental study Gai11JCP , here we provide
theoretical arguments and simulations to quantify the dependence of enhanced
stabilities of proteins, expressed in terms of the $\Phi_{C}$-dependence of
the melting temperatures, $T_{m}(\Phi_{C})$, in the presence of spherical
crowding agents. Although the use of spherical crowding agents is a caricature
of the cellular environment and perhaps even model agents used in in vitro
studies, a quantitative treatment of the simpler system is a useful first step
towards a more realistic description of the heterogeneous milieu containing
polydisperse crowding agents with a variety of shapes. We show that the
increase in $T_{m}(\Phi_{C})$, relative to the value in the absence of
crowding, increases as $\Phi_{c}^{\alpha}$ where the value of the effective
exponent, $\alpha$, is determined by the characteristics of the unfolded
states in the absence of crowding agents ($\Phi_{C}=0$). The theoretical
arguments lead to an upper and lower bound for $\alpha$ but the precise
requires a fit to the experimental data. We validate the predictions using
coarse-grained simulations Hyeon11NatComm  of $\lambda_{6-85}$ repressor in
the presence of spherical crowding particles. The values of $\alpha$ extracted
from simulations and experiments are consistent with the predicted bounds. We
also examine the plausible effects of the non-specific soft attractions
between the crowding particles and the polypeptide chain in order to
understand the very modest (less than $k_{B}T$) destabilization found in the
presence of cell lysates. We argue that the thermodynamic stability of
proteins even in complex environment is largely determined by excluded volume
interactions. However, as predicted theoretically, the effects of steric
interactions on the stability could be negligible (practically no effect), and
is determined by the ratio of the size of the unfolded state of the protein
when $\Phi_{C}=0$ to the size of largest crowders in a soup containing mixture
of crowding agents.

## Theory

Melting temperature increases with $\Phi_{C}$: The physical arguments hinge on
the observation that when $\Phi_{C}\neq 0$ the polypeptide chain would prefer
to be localized in a region that is devoid of hard sphere-like crowding
particles. On general theoretical grounds Edwards88JCP ; Thirum88PRA ;
vandershoot98Macro , it can be shown that the probability of finding a void
that is large enough to accommodate a polypeptide chain with radius of
gyration $R_{g}$ decreases exponentially as $\Phi_{C}$ increases. However, the
fluctuations in density of the crowding particles would create a void whose
optimal size $D$ (Fig. 1) is likely to be spherical Honeycutt89JCP . In this
sense the effects of crowding can be approximately mimicked by confining the
polypeptide chain to a spherical cavity of size $D$ Cheung05PNAS . It is
straightforward to show that

$D=R_{c}\Phi_{C}^{-{\frac{1}{3}}}$ (1)

where $R_{c}$ is the radius of the spherical crowding particle. Assuming that
the mapping between crowding and confinement is reliable, as appears to be the
case based on simulations Cheung05PNAS , we can estimate the decrease in free
energy of the unfolded state due to localization of the polypeptide chain in a
spherical cavity as, $\Delta F_{U}(\Phi_{c})\approx
k_{B}T(R_{g}/D)^{1/{\nu}}$, where $\nu$ is the Flory exponent describing
$R_{g}\approx aN^{\nu}$, $k_{B}$ and $T$ are the Boltzmann’s constant and
temperature respectively, and $N$ is the number of amino acid residues, and
$a$ is approximately the distance between $C_{\alpha}$ atoms. In using this
estimate for $\Delta F_{U}$, we assumed that the unfolded state can be treated
as a self-avoiding walk. A more refined treatment which takes into account
intra polypeptide interactions GrosbergBook ; Thirumalai03PNAS , always
present even in high denaturant concentration, shows that

$\Delta F_{U}=k_{B}T(R_{g}/D)^{\frac{3}{(3\nu-1)}}.$ (2)

Assuming that the folded state is not significantly affected by the crowding
particles, we obtain from Eqs. (1 and 2) the following equation for $\Delta
T_{m}(\Phi_{C})=T_{m}(\Phi_{C})-T_{m}(0)$,

$\Delta
T_{m}(\Phi_{C})\approx\left(\frac{\Phi_{C}}{R_{C}^{3}}\right)^{\alpha}$ (3)

where $\alpha=\frac{1}{(3\nu_{eff}-1)}$. Because of both finite-size of
proteins as well as the presence of intra polypeptide attractive interactions
(at least between hydrophobic residues) $\nu$ in Eq. (3) should be treated as
an effective exponent ($\nu_{eff}$), and could depend on the protein of
interest. However, the upper bound for $\nu_{eff}$ is $\approx$ 0.6. The lower
bound, at which the folding and collapse temperatures are nearly coincident
for single domain proteins Camacho93PNAS , is $\approx$ 0.5. Thus, the lower
and upper bounds on $\alpha=\frac{1}{(3\nu_{eff}-1)}$ are 1.25 and 2,
respectively. If $\alpha<$ 2, it implies that there is residual structure in
the unfolded state. The $\alpha$-exponent in Eq. (3) is determined by the
dimensions of proteins in the unfolded state, and should be valid for any
protein provided only interactions between crowders and polypeptide chain are
relevant.

Stability changes as a function of $\Phi_{C}$ and $R_{C}$: In order provide a
molecular picture of the stability changes of the protein, which can be
computed in simulations, we consider a polymer chain with radius of gyration
$R_{\rm g}$ in a suspension of crowders with density $\rho_{\rm C}$. The
volume fraction of the suspension $\Phi_{\rm C}$ scales as $\rho_{\rm C}R_{\rm
C}^{3}$. When the polymer comes in contact with a crowding particle, the
polymer segments become depleted (AO effect) from the particle surface,
resulting in an increase in the polymer free energy. If the crowding particles
are much smaller than the polymer, $R_{\rm C}\ll R_{\rm g}$, the free energy
increase per particle is given by Odijk00PhysicaA

$\frac{\Delta F_{1}}{k_{\rm B}T}\sim\frac{R_{\rm C}}{R_{\rm g}}$ (4)

for ideal polymers, and by

$\frac{\Delta F_{1}}{k_{\rm B}T}\sim\left(\frac{R_{\rm C}}{R_{\rm
g}}\right)^{4/3}$ (5)

for polymers in good solvent deGennesbook . We can rewrite both the results as

$\frac{\Delta F_{1}}{k_{\rm B}T}\sim\left(\frac{R_{\rm C}}{R_{\rm
g}}\right)^{3-1/\nu},$ (6)

where $\nu$ is the Flory exponent, $R_{\rm g}\sim aN^{\nu}$, $N$ is the number
of segments in the polymer and $a$ is the segment size.

Suppose that the crowders, which have been expelled from the polymer interior,
stay close to the polymer surface (this assumption is justified below). Then,
the number of crowders in contact with the polymer, $n$, can be estimated as

$n\sim\rho_{\rm C}R_{\rm g}^{3}\sim\Phi_{\rm C}\left(\frac{R_{\rm g}}{R_{\rm
C}}\right)^{3},$ (7)

so that the total free energy loss due to crowding, $\Delta F$, is

$\frac{\Delta F}{k_{\rm B}T}=n\frac{\Delta F_{1}}{k_{\rm B}T}\sim\Phi_{\rm
C}\left(\frac{R_{\rm g}}{R_{\rm C}}\right)^{1/\nu}.$ (8)

In the limit $\Phi_{\rm C}\to 1$, Eq. 8 becomes the well known result for the
free energy of confining a polymer in a slit-like cavity of size $R_{\rm C}$.
It is important to remember that both Eq. 6 and Eq. 8 are valid only in the
limit $R_{\rm C}/R_{\rm g}\to 0$.

For macromolecular crowding in a cellular environment, the relevant regime is
$R_{\rm C}\sim R_{\rm g}$. In this case, we have $\Delta F_{1}\sim k_{\rm B}T$
in Eq. 6 and

$\frac{\Delta F}{k_{\rm B}T}\sim n$ (9)

in Eq. 8. Therefore, we anticipate that the free energy of the polymer will
increase linearly with the number of crowders at the polymer surface, $n$.
Furthermore, for $R_{\rm C}\sim R_{\rm g}$, the volume excluded to the
crowders by the polymer, $V_{\rm exc}$, is not well approximated by $R_{\rm
g}^{3}$ as was done in Eq. 7. The actual value of $V_{\rm exc}$ will increase
sharply with the crowder size $R_{\rm C}$, yielding a weaker dependence of $n$
on $R_{\rm C}$ than $R_{\rm C}^{-3}$ in Eq. 7.

## Simulations

In order to test the theoretical predictions, we performed coarse-grained (CG)
simulations of $\lambda_{6-85}$ repressor (see Methods for details) whose
folding under infinite dilution conditions and in vivo have been
experimentally investigated Ghaemmaghami01NSB ; Prigozhin11JACS . We assume
that the analysis presented above applies to the unfolded state of the
protein, whereas the folded state is not significantly affected by the
crowders. Then, Eq. 9 determines the increase in the stability of the protein
due to crowding. A relationship similar to Eq. 9 can also be written for the
increase in the melting temperature of the protein, $\Delta T_{\rm m}$, as
long as the crowding effect is a relatively small perturbation. The use of
$C_{\alpha}-SCM$ is fully justified here because our theoretical arguments
suggest that the predictions for $\Delta T_{m}(\Phi_{C})$ are universal
depending only globally on the characteristics of the unfolded state. In a
number of applications, we have shown that simulations based on
$C_{\alpha}-SCM$ captures the folding reactions accurately Klimov00PNAS ;
OBrien08PNAS ; Reddy12PNAS .

The peak in the temperature dependence of the heat capacity (upper left corner
of Fig. 2) is associated with the melting temperatures. The shift in the
melting temperature, $\Delta T_{m}(\Phi_{C})$, shown in green circles in Fig.
2, can be fit extremely well using Eq. (3) with $\alpha\approx$ 1.5 (black
line in Fig. 2), which is within the predicted bounds. The effective exponent
$\nu_{eff}\approx$ 0.55. Our previous simulation results for effect of
spherical crowding particles on the WW domain showed that $\Delta
T_{m}(\Phi_{C})\approx\Phi_{C}^{1.8}$ leading to $\nu_{eff}\approx$ 0.52. For
both these systems the predicted bounds are satisfied further justifying the
validity of the entropic stabilization mechanism. More importantly, theory and
simulations in Fig. 2 show that $\Delta T_{m}(\Phi_{C}=0.15)$ is $\approx$3∘,
which is very good agreement with experiments (experimental value for $\Delta
T_{m}(\Phi_{C}=0.15)$ is $\approx$4∘ Denos12FaradDisc  in which Ficoll was
used as a crowding agent. We find the agreement particularly satisfying
because we did not adjust any parameter to fit the experimental data.

The simulation data confirm that the melting temperature of $\lambda_{6-85}$
increases linearly with the number of crowders localized at the protein
surface, $n$ (Fig. 3a). The increase in the melting temperature is
approximately 1 ∘C per crowder. For given $\Phi_{\rm C}$ and $R_{\rm C}$, the
value of $n$ was obtained by directly counting the number of crowders whose
centers were within distance $4R_{\rm C}/3$ from the protein surface. Each
reported value represents an average over 10,000 protein conformations in the
unfolded state at 105 ∘C. We find that $n$ follows the power law,

$n\propto\left(\frac{\Phi_{\rm C}}{R_{\rm C}}\right)^{1.43},$ (10)

for all considered combinations of $\Phi_{\rm C}$ and $R_{\rm C}$ (3b), and
hence $\Delta T_{\rm m}\propto\left(\frac{\Phi_{\rm C}}{R_{\rm
C}}\right)^{\alpha}$, where $\alpha=1.43$.

In Fig. 3c, the dependence of $n$ on $\Phi_{\rm C}$ at $R_{\rm C}=24$ Å (red
symbols) is compared to the estimate $\rho_{\rm C}V_{\rm exc}$ (pink symbols),
where $V_{\rm exc}$ was computed numerically for the same set of 10,000
simulation snapshots that were used to determine $n$. Although the estimate
$\rho_{\rm C}V_{\rm exc}$ is found to be fairly accurate, it specifies a
linear dependence of $n$ on $\Phi_{\rm C}$ as opposed to the observed
nonlinear dependence (3c). These results indicate that inter-particle
correlations in crowding suspensions promote clustering of crowders around the
protein surface.

Fig. 3d shows $n(R_{\rm C})$ at $\Phi_{\rm C}=0.25$, as obtained by direct
counting (green symbols) or from the estimate $\rho_{\rm C}V_{\rm exc}$ (pink
symbols). Both curves decay more slowly than $R_{\rm C}^{-3}$ specified in Eq.
7, illustrating a large contribution of the crowder size $R_{\rm C}$ to the
excluded volume $V_{\rm exc}$.

We conclude that the value of $\alpha$ in Eq. 3 is determined by crowder-
protein and crowder-crowder correlations and as such it cannot be expressed in
terms of the polymer scaling exponents alone although the precise value of
$\alpha$ extracted from simulation satisfies the expected bounds. Thus,
experiments can be analyzed using Eq. 3 using a single adjustable parameter,
the effective exponent $\alpha$.

## Comparison with experiments

Waegle and GaiGai11JCP  found empirically that the melting temperature of a
76-residue protein Ubiquitin(Ub) increases algebraically as a function of
$\Phi_{C}$ Cheung05PNAS  and is well fit using Eq.(3). They used dextran of
differing molecular weights and Ficoll 70 as crowding agents. The $\Phi_{C}$
dependent shift in the melting temperature was obtained from thermal melting
curves inferred from Fourier transform infrared measurements. The measurements
of $\Delta T_{m}(\Phi_{C})$ as a function of $\Phi_{C}$ confirm the
theoretical prediction with $\alpha$ ranging from $1.4\leq\alpha\leq 2.1\pm
0.1$. This implies that $0.5\leq\nu_{eff}\leq 0.6$, which brackets the
predicted theoretical bound. Interestingly, $\nu_{eff}$ =0.5 is obtained for
high molecular particles Dextran40 and Dextran60. The finding that $\nu_{eff}$
= 0.5 implies that the crowding particles poises the solvent to be close to
the $\Theta$-point.

It is more difficult to compare the dependence of $\Delta T_{m}(\Phi_{C})$ on
the crowder size because it is likely that the crowding particles considered
in Gai11JCP  are not spherical. Nevertheless, the observed weak dependence of
$\Delta T_{m}(\Phi_{C})$ on $R_{C}$ is qualitatively consistent with the
theoretical predictions. It can be shown that the maximum effect of crowding
arises when $R_{C}\leq R_{g}$. When $R_{g}\geq R_{C}$, the stabilities become
independent of $R_{C}$ because in this limit the polypeptide chain behaves as
though it is trapped between slits. In the case of Ub Gai11JCP , the ratio
$\frac{R_{g}}{R_{C}}$ is largest for Dextran 6 and is less than unity for
higher molecular higher weight Dextran. For Ficoll 70, $\nu_{eff}\approx$ 0.5
based on $\alpha\approx$ 2.0, which also suggests that
$\frac{R_{g}}{R_{C}}<1$. Therefore, it is not surprising that there is only a
weak dependence of $\Delta T_{m}(\Phi_{C})$ on the size of the crowding
particles. Our theory provides a quantitative explanation of the experiments
on Ub.

## Role of non-specific attractions

A few experiments have reported that proteins can be destabilized (mildly)
under conditions that apparently mimic cellular conditions. Based on these
findings it has been concluded that the predictions of the crowding theory,
which only consider steric interactions, must be incorrect. We first provide a
theoretical argument showing that, even in the presence of non-specific
attractive interactions between crowding particle and a polypeptide chain in a
cellular environment, the macromolecular interactions are likely to be
dominated by excluded volume effects. Let $u(r)$ be the interaction potential
between residues in a protein (or RNA) and macromolecular crowders, such that
$u(r)$ has an excluded volume part and an attractive part. The relative
strength of the two contributions in $u(r)$ can be assessed using the second
virial coefficient $B_{2}$,

$B_{2}=\frac{1}{2}\int_{0}^{\infty}\left[1-\exp\left(-\frac{u(r)}{k_{\rm
B}T}\right)\right],$ (11)

where $T$ is the temperature and $k_{\rm B}$ is the Boltzmann constant. The
second virial coefficient vanishes if the attractive interaction exactly
balances the excluded volume contribution. To derive an analytical expression
for $B_{2}$, we assume that $u(r)$ is a square-well potential comprising of
both hard and soft interactions (Fig. 4),

$\displaystyle u(r)$ $\displaystyle=$ $\displaystyle\infty,r<D,$
$\displaystyle u(r)$ $\displaystyle=$ $\displaystyle-\varepsilon,D\leq r\leq
D+\sigma,$ $\displaystyle u(r)$ $\displaystyle=$ $\displaystyle 0,r>D+\sigma,$
(12)

where $D$ is the contact distance, corresponding to the universally relevant
excluded volume, and $\sigma$ measures the range of the non-specific
attractions. In the case of a protein interacting with other crowding
proteins, we estimate $D=R_{\rm a}+R_{\rm C}$ and $\sigma=2R_{\rm a}$, where
$R_{\rm a}$ and $R_{\rm C}$ are the radii of the amino acid and crowder
respectively. From Eqs. (11) and (12) we find

$\frac{3B_{2}}{2\pi
D^{3}}=\left(1+\frac{\sigma}{D}\right)^{3}-\exp\left(\frac{\varepsilon}{k_{\rm
B}T}\right)\left[\left(1+\frac{\sigma}{D}\right)^{3}-1\right].$ (13)

The strength of non-specific attractions that is sufficient to neutralize the
excluded volume effects, $\varepsilon_{0}$, follows by setting $B_{2}=0$ in
Eq. (13) yielding,

$\frac{\varepsilon_{0}}{k_{\rm
B}T}=-\ln\left[1-\left(1+\frac{\sigma}{D}\right)^{-3}\right].$ (14)

For any other form of the interaction potential $u(r)$ that is short-ranged,
the attraction strength $\varepsilon_{0}$ will also depend only on the ratio
of length scales $D$ and $\sigma$. The value of $\varepsilon_{0}$ increases
sharply with decreasing $\sigma$ (Fig. 4b), indicating that strong attractive
interactions are required for complete neutralization of the excluded volume
effects when $\sigma/D<1$. We note that the condition $\sigma/D<1$ is
satisfied for all realistic values of $R_{\rm a}$ and $R_{\rm C}$.

Setting $R_{\rm C}=3R_{\rm a}$ to model a crowder protein of a minimum size,
we find $\varepsilon_{0}/k_{\rm B}T\approx 0.35$. This value is
unrealistically large for any non-specific attractions that may be present
between macromolecules. To illustrate this point, we observe that most
biomolecules melt at temperatures that are only insignificantly higher (on the
scale of $k_{\rm B}T$) than the physiological temperature. Assuming that the
second virial coefficient of interactions between two amino acids vanishes at
the melting temperature, we can use Eq. (14), with $D=\sigma$, to estimate the
strength of non-specific attractions between amino acids. We find
$\varepsilon_{0}/k_{\rm B}T\approx 0.13$, which is of the order of the
interaction strength between two carbon atoms, and approximately 3 times
weaker than necessary to negate the crowder-amino acid excluded volume effect.
Thus, for any realistic $R_{\rm C}$, $R_{\rm a}$ and $\varepsilon_{0}$ the
value of $B_{2}$ is positive and likely dominated by steric interactions.

There are additional arguments which put an upper limit on the strength of
non-specific attractions between macromolecules in the cell and justify the
positive values of $B_{2}$. First, in the case $B_{2}<0$, macromolecular
crowders can adsorb onto an unfolded protein and interfere with its folding.
(Such unwarranted interactions may be prevented by ATP-consuming chaperones,
which are not germane to the experiments discussed below.) Furthermore,
suspensions of macromolecular crowders with sufficiently negative $B_{2}$ and
volume fractions representative of those in the cell (0.2–0.4) could form
crystalline phases. Clearly, these physical phenomena do not satisfy the
conditions of non-specificity. We therefore conclude that, even if non-
specific attractions are present between macromolecules in a cellular
environment, their effect must be small compared to steric interactions.

Analyses of experiments - Numbers matter: In light of the arguments above how
can one understand the claims that in vivo environment or cell lysates
destabilize proteins? This can only be done by examining the experimental
results quantitatively using an analytic theory that captures one limiting
case quantitatively, as we have done here. It is useful to remind the readers
that crowding theory (the one which examines only the consequences of steric
interactions) predicts that the native state would be stabilized with respect
to the $\Phi_{C}=0$ situation, resulting in negative $\Delta\Delta G=\Delta
G(\Phi_{C})-\Delta G(0)<0$ where $\Delta G$s are free energy differences
between the folded and unfolded states. According to crowding theory, based on
the concept of depletion interaction, the magnitude of $\Delta\Delta G$
depends on the ratio $\frac{R_{g}(0)}{R_{\rm C}}$, and is negligible when
$\frac{R_{g}(0)}{R_{\rm C}}$ is less than unity.

A cell lysate presumably contains macromolecules of differing sizes (a
polydisperse mixture), which makes it non-obvious on how one ought to choose
$R_{\rm C}$. Our theory and related works Minton05BJ , rooted in the concept
of depletion forces, predicts that the polypeptide chain in a polydisperse
soup of macromolecules is most likely surrounded by large-sized crowders
Shaw91PRA  in order to maximize the total entropy of the system Denesyuk11JACS
; Kang15JACS . In a cell lysate, we expect this be particles like ribosomes or
other protein complexes with $R_{\rm C}\approx$ 10 nm, which is the
approximate size of a ribosome. Thus, significant entropic stabilization is
possible only if $R_{g}(0)\approx$ 10 nm. With this simple description of the
theory, we analyze experiments examining crowding effects on the stabilities
of CI2 ($N_{aa}=64;R_{g}(0)\approx 2.4nm$), $\lambda_{6-85}$
($N_{aa}=80;R_{g}(0)\approx 2.8nm$), CRABP ($N_{aa}=136;R_{g}(0)\approx
3.8nm$), and VlSE ($N_{aa}=341;R_{g}(0)\approx 6.6nm$), where $N_{aa}$ is the
number of amino acids and the radius of gyration of the unfolded state in the
absence of crowders is obtained using $R_{g}(0)\approx 0.2N_{aa}^{\nu}$nm with
$\nu\approx 0.6$. Because the $R_{g}(0)$ values are not large compared to the
size expected for significant stabilization, crowding theory would predict
that $\Delta\Delta G$ to be negligible at most values of $\Phi_{c}$ of
interest.

The reported values of $\Delta\Delta G$s for these proteins in a cell lysates
or cell-like environment are (in units of kcal/mol) -0.6$\pm$0.1 Sarkar13PNAS
, $\approx 0$, -0.2, and -0.5 for CI2, $\lambda_{6-85}$ Tai16FEBSLett , CRABP
I (cellular retinoic acid-binding protein I) Ignatova04PNAS , and VlsE
Guzman14JMB  respectively. These values are small (less than $k_{B}T$). In
three cases they suggest destabilization of proteins. Unless there are cases
where the extent of destabilization is greater, we interpret that these
results are not inconsistent with crowding theory in contrast to the
advertisement in the experimental papers. We believe that the onus is on the
experimentalists to produce examples where the extent of destabilization
exceeds $k_{B}T$. We hasten to add that destabilization of proteins is not
ruled out if non-specific attraction is strong enough (see Eq. 14). However,
the currently available examples are not the best ones to illustrate the
extent of destabilization in cells. From these experiments one can infer that
cell lysates do not significantly the stabilities of proteins, which can be
accommodated using the crowding theory based on the idea of depletion
interaction.

## Conclusions

If the volume excluded to the proteins by the crowders is the predominant
interaction then the complicated problem of protein stability, expressed using
the dependence of the melting temperature on $\Phi_{C}$ is accurately
predicted using Eq. (3), thus providing a firm theoretical framework. Surely,
this is an interesting limiting case. Remarkably, the exponent $\alpha$ is
predicted to be universal and depends only on the properties of the unfolded
state at $\Phi_{C}$ = 0. A few additional remarks are worth making. (1) The
effects of excluded volume interactions, at a given $\Phi_{C}$, should give an
upper bound to the enhancement in protein stability. Other favorable
interactions, which alter the enthalpy of interactions, could diminish the
stability of proteins, thus neutralizing the effect of steric interactions
both on protein stability Ghaemmaghami01NSB  and association between proteins
Rosen11JPCB . (2) The theoretical predictions made here require that the
polymer fluctuations in the unfolded state determine the dependence of $\Delta
T_{m}(\Phi_{C})$ on $\Phi_{C}$. Hence, ideas based on Scaled Particle Theory,
which can capture some aspects of crowding, cannot be used to describe many
features described here and elsewhere Denesyuk11JACS ; Cheung05PNAS  even when
non-specific attractions could be neglected. (iii) In analyzing experiments
and simulations the exponent $\alpha$ should be chosen to fit the experimental
data. The demonstration that for $\lambda_{6-85}$ the value of $\alpha$ cannot
be tidily predicted from well-known polymer scaling exponents show that it
should be considered to be a fit parameter. The bounds on $\alpha$ follow from
polymer scaling exponents. (iv) In order to establish that realistic soup of
cellular milieu (a multi component system) destabilize proteins two conditions
must be met. First, the extent of destabilization must exceed $k_{B}T$.
Second, under these conditions it must be demonstrated that the native state
has not been altered. It may be the case that there are changes in the
structures of the folded states and intermediates in cellular environment
(macromolecules and various osmolytes) as atomic detailed simulations
Feig17JPCB  seem to suggest.

Polydisperse Effects: Perhaps, the most interesting aspect of the theory,
based on the entropic stabilization of the folded state, is its utility in
providing an alternate explanation for the apparent near universal
destabilization of proteins in cell-like environments. It should be emphasized
that theory does not preclude the possibility that crowders destabilize
proteins. Two scenarios could be envisaged. (I) The popular explanation is
that soft interactions between crowders and proteins Cheung06JMB ;
Drishti19ChemRev ; Sarkar13PNAS ; Sapir15CurrOpinColl ; Sapir14JPCL ;
Danielsson15PNAS ) result in negation of the stabilizing effect due to volume
exclusion. For this to occur the interaction strength has to be substantial
(see Eq.14 and Rosen11JPCB  for an estimate for a protein complex). We had
previously shown that under these conditions the native state may not be
stable and other states, including the possibility that the protein would
weakly adsorb onto the crowders Cheung06JMB , are populated with higher
probability. Under these conditions, which apparently is realized in atomic
simulations of folding in the presence of crowders, comparing of the free
energy changes in the crowded milieu is not meaningful. (II) It is also
possible that crowders, interacting with the protein solely through excluded
volume interactions, largely affect only the unfolded state without affecting
the folded state. In this scenario, crowders would render the unfolded state
compact, as predicted theoretically Edwards88JCP ; Thirum88PRA , facilitating
intra peptide attraction. This would stabilize the unfolded state
enthalpically increase the barrier to folding. As a consequence, the stability
would decrease and the folding time is predicted to be slower. This scenario
explains the modest changes in stability and folding time for VLSE and PGK in
cells relative to in vitro (see Table 1 in Tai16FEBSLett ). (III) The modest
destabilization observed in several experiments is usually rationalized in
terms of weak interactions, which of course is a possibility that cannot be
ruled out. Based on the simulations on DNA flexibility Kang15JACS  and
theoretical considerations Kang15PRL , we believe that the concept of
depletion forces when extended to polydisperse crowding agents offer an
alternative explanation. Consider a mixture of crowding agents with different
sizes and shapes, which might be a better mimic of the cytoplasm. The AO
picture predicts that maximization of entropy is realized if the protein is
surrounded by the largest crowding particles. If this were the case then the
stability would not be altered significantly Denesyuk11JACS , which would not
be inconsistent with experimental data analyzed here. A potential in vitro
experiment which could shed light on this issue is to use Ficoll and Dextran
of various sizes (with some crowding particles exceeding the radius of
gyration of the unfolded states), and assess the changes in the melting
temperatures.

###  .1 Methods

Model: The polypeptide chain is simulated using a coarse-grained model, in
which each amino acid is replaced by two spherical beads, representing a
$C_{\alpha}$ atom and a side chain ($SC$) Klimov00PNAS . The use of
$C_{\alpha}-SC$ model is fully justified here because our theory suggests that
the predictions for $\Delta T_{m}(\Phi_{C})$ and the associated stabilities
are universal depending globally only on the characteristics of the unfolded
state. The energy function in the $C_{\alpha}-SC$ coarse-grained model,
$U_{\rm CG}$, has the following four components,

$U_{\rm CG}=U_{\rm CC}+U_{\rm SC}+U_{\rm EV}+U_{\rm NAT},$ (15)

corresponding to $C_{\alpha}-C_{\alpha}$ and $C_{\alpha}-SC$ bond length
constraints, excluded volume repulsions and native interactions. Bond lengths
are constrained by harmonic potentials, $U_{\rm CC}(\rho)=k(\rho-\rho_{\rm
CC})^{2}$ and $U_{\rm SC}(\rho)=k(\rho-\rho_{\rm SC})^{2}$, where $k=30$ kcal
mol-1Å-2. The equilibrium distance between two $C_{\alpha}$ atoms, $\rho_{\rm
CC}$, is defined individually for each bond as the corresponding
$C_{\alpha}$-$C_{\alpha}$ distance in the protein crystal structure (PDB code
1LMB). Similarly, $\rho_{\rm SC}$ is the distance between each residue’s side
chain center of mass and $C_{\alpha}$ atom in the PDB structure.

We model the excluded volume interactions between two coarse-grained beads $i$
and $j$ separated by distance $r$ using a generalized Weeks-Chandler-Andersen
(WCA) potential,

$\displaystyle U_{\rm EV}(r)$ $\displaystyle=$ $\displaystyle\varepsilon_{\rm
EV}\frac{{\rm min}(D_{i},D_{j})}{D_{\rm C}}\left[\left(\frac{D_{\rm
C}}{r+D_{\rm C}-D_{ij}}\right)^{12}-2\left(\frac{D_{\rm C}}{r+D_{\rm
C}-D_{ij}}\right)^{6}+1\right],\ r\leq D_{ij},$ $\displaystyle U_{\rm EV}(r)$
$\displaystyle=$ $\displaystyle 0,\ r>D_{ij},$ (16)

where ${\rm min}(D_{i},D_{j})$ is the smaller of the bead diameters $D_{i}$
and $D_{j}$, $D_{ij}=0.5(D_{i}+D_{j})$ and $\varepsilon_{\rm EV}=1$ kcal/mol.
The diameter of a bead representing a $C_{\alpha}$-atom is $D_{\rm C}=3.8$Å.
For each residue, the value of $D_{\rm S}$ is given by $V_{\rm S}=\pi D_{\rm
S}^{3}/6$, where $V_{\rm S}$ is the van der Waals volume of the side chain
computed from the PDB coordinates of its individual atoms using the AMBER94
atomistic van der Waals radii. Thus, side chains of the same kind can have
somewhat different $D_{\rm S}$ depending on their configuration in the PDB
structure. Interactions of crowders with the protein residues and other
crowders are modeled using the same potential as in Eq. (16). We chose the
generalized form of the WCA potential because it is well suited to model
excluded volume between two particles with very different diameters, such as a
$C_{\alpha}$-atom and a macromolecular crowder. The ratio ${\rm
min}(D_{i},D_{j})/D_{\rm C}$ in Eq. (16) rescales the interaction strength
$\varepsilon_{\rm EV}$ in proportion to the surface contact area between
particles $i$ and $j$.

Native interactions between protein coarse-grained beads are modeled by a
Lennard-Jones potential,

$\displaystyle U_{\rm NAT}(r)=\varepsilon_{\rm
NAT}\left[\left(\frac{r_{0}}{r}\right)^{12}-2\left(\frac{r_{0}}{r}\right)^{6}\right],$
(17)

where $r_{0}$ is the corresponding interbead distance in the coarse-grained
PDB structure. The native interactions are defined only for those pairs of
beads for which $r_{0}<8$ Å. The value of $\varepsilon_{\rm NAT}$ is the same
for all native interactions, $\varepsilon_{\rm NAT}=0.43$ kcal/mol, and has
been adjusted so that the experimental melting temperature of $\lambda_{6-85}$
is reproduced in simulation ($T=56^{\circ}$C).

Equations of motion: The protein and crowder dynamics are simulated by solving
the Langevin equation, which for bead $i$ is
$m_{i}\ddot{\mathbf{r}}_{i}=-\gamma_{i}\dot{\mathbf{r}}_{i}+\mathbf{F}_{i}+\mathbf{f}_{i}$,
where $m_{i}$ is the bead mass, $\gamma_{i}$ is the drag coefficient,
$\mathbf{F}_{i}$ is the conservative force, and $\mathbf{f}_{i}$ is the
Gaussian random force,
$\left<\mathbf{f}_{i}(t)\mathbf{f}_{j}(t^{\prime})\right>=6k_{\rm
B}T\gamma_{i}\delta_{ij}\delta(t-t^{\prime})$. The mass of $C_{\alpha}$ bead
is the molecular weight of a carbon atom and the mass of a $SC$ bead is the
total molecular weight of the corresponding side chain. We use the mass of 8.6
kDa for a crowder with diameter 24 Å, which is consistent with typical protein
densities. The mass of any other crowder $i$ with diameter $D_{i}$ is
estimated as $m_{i}=8.6(D_{i}/24)^{3}$ kDa. The drag coefficient $\gamma_{i}$
is given by the Stokes formula, $\gamma_{i}=3\pi\eta D_{i}$, where $\eta$ is
the viscosity of the medium. To enhance conformational sampling
Honeycutt92Biopolymers , we take $\eta=10^{-5}$Pa$\cdot$s, which equals
approximately 1% of the viscosity of water. The Langevin equation is
integrated using the leap-frog algorithm with a time step $\Delta t=10$ fs.

Acknowledgments: This work was done while the authors were in the Institute
for Physical Sciences and Technology at the University of Maryland. We
appreciate the useful comments from Martin Grubele and Ben Schuler. We are
grateful to the National Science Foundation (CHE 19-00093) and the Collie-
Welch Chair (F-0019) for supporting this research.

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Figure Captions

Figure 1 Snapshot of the encapsulated $\lambda$-repressor from coarse-grained
SOP simulations in the presence of crowding particles shown in dark blue. The
protein is localized in a roughly spherical region with diameter $D$. This
picture provides a physical basis for approximate mapping between crowding and
confinement.

Figure 2 Dependence of the shift in the melting temperature ($\Delta
T_{m}(\Phi_{C})$ for $\lambda$-repressor as a function of $\Phi_{C}$. The
green circles are obtained from simulations and the black line is a
theoretical fit using Eq. (3) with $\alpha$ = 1.5. The structure on the right
is a ribbon representation of the native state of the protein. The melting
temperatures are associated with the peaks in the heat capacity such as the
ones shown for $\Phi_{C}$ = 0 (black curve) and $\Phi_{C}$ = 0.3 (dashed red
curve).

Figure 3 The increase in the melting temperature, $\Delta T_{\rm m}$, and the
number of crowders localized at the protein surface, $n$, obtained in
simulations of $\lambda$ repressor. The parameter $n$ is defined as the number
of crowders positioned within distance $4R_{\rm C}/3$ from the protein
surface. Red squares show simulation results for $R_{\rm C}=24$ Å and
$\Phi_{\rm C}=0.05$, 0.1, 0.15, 0.2, 0.25, 0.3. Green circles are for
$\Phi_{\rm C}=0.25$ and $R_{\rm C}=12$, 20.5, 24, 41, 48, 96 Å. Pink symbols
in (c) and (d) show $n$ as estimated by $\rho_{\rm C}V_{\rm exc}$, where
$V_{\rm exc}$ is the protein-crowder excluded volume computed numerically from
simulation snapshots. The solid lines correspond to functions (a) $y=1.08x$,
(b) $y=3452x^{1.43}$, (c) $y=36.5x^{1.43}$, and (d) $y=479x^{-1.43}$.

Figure 4 (a) Square-well potential, $u(r)$, used in the calculation of the
second virial coefficient, $B_{2}$. (b) $\varepsilon_{0}$ as a function of
$\sigma/D$. By definition $B_{2}$ vanishes at $\varepsilon=\varepsilon_{0}$.

Figure 1:

Figure 2:

Figure 3:  Figure 4: