Macromolecular condensation buffers
intracellular water potential
In the format provided by the
authors and unedited
Nature | www.nature.com/nature
Supplementary information
https://doi.org/10.1038/s41586-023-06626-z
Supplementary discussion
Chapter I/ Fitting of osmotic potential curves
As may be appreciated from Extended Data Fig.1b,d
and f, the relationship between osmotic
potential and concentration of various biologically important macromolecules departs markedly from
linearity. The point of this supplementary discussion is to establish an empirical effective measure for
quantifying
departure from linearity, so that different solutes and temperatures may be compared.
PEG offers a good avenue to evaluate how departure from linearity varies as a function of
physiological parameters, such as temperature, solvent composition (D
2
0 or H
2
O) and polymer size,
since
PEG has no solubility limit
and
does not phase separate across physiological temperatures (Fig.
4a)
.
The osmotic potential of dilute solutions is classically modelled by
van’t Hoff’s law (equation
1), with
푖푖
the
v an’t Hoff
factor,
푅푅
the gas constant, and
푇푇
the temperature in Kelvins:
−훹훹
휋휋
(
퐶퐶
,
푇푇
)
=
푖푖
푅푅푇푇퐶퐶
(1)
I
n the more general case,
−훹훹
휋휋
is fit to the polynomial equation 2, where α & β are termed the
first and second virial coefficients, respectively
1–3
:
−훹훹
휋휋
(
퐶퐶
,
푇푇
) =
푖푖
푅푅푇푇퐶퐶
(
1 +
훼훼
퐶퐶
+
훽훽퐶퐶
2
+
⋯
)
(2)
No
te that equation (2) converges to equation (1) in dilute solutions, where molecules are
sparsely distributed, and that equation (2) departs markedly from linearity at high solute
concentrations, where the
퐶퐶
2
and
퐶퐶
3
components
become dominant. At first, we thus attempted to
use equation 2 to model our osmotic potential curves in order to employ the value of the virial
coefficients as an empirical measure of departure from linearity (i.e. the higher the value of the virial
coefficients, the larger the departure from linearity). For this, we constrained the values of the viral
coefficient to be positive.
A priori
, is appears plausible that the virial coefficients might vary as a consistent function of
solute identity and temperature. Note, however, that a modest temperature 10K decrease, from 310K
to 300K elicits a two
-fold increase in osmotic potential of BSA (Fig.1a). This is already challenging to
reconcile with equation 2, which is linear with respect to temperature (in Kelvin) and instead predicts
a ~3% decrease in osmotic potential with temperature decrease, rather than the observed increase.
As can be appreciated in Supplementary Figure 1, the osmotic potential of PEG 35KDa
solutions as a function of concentration is poorly fit by equation (1) at 290K, and better described by
equation (2). For this latter, note that the higher the order of the
polynomial (i.e. the more virial
coefficients allowed in the model), the better the fit.
By extensively and systematically measuring the osmotic potential of PEG solutions for
different PEG sizes and different temperature (raw data in Extended Data Fig.1d),
the variation in the
values of the
virial coefficients as a function of these parameters can be determined by simultaneously
by fitting all datasets to equation 2 (
see Supplementary Figure 2)
. We performed this analysis by
considering the development of Equation 2 to the 2nd or 3rd order (that is, considering 1st or 2nd
virial coefficients).
As can be observed in Supplementary Figure 2, there is no clear trend of the value of the virial
coefficients as a function of PEG size nor temperature. If equation 2 was valid to model the interaction
of PEG with the solvent in the ranges considered in our experiments, the virial coefficients would be
expected to show consistent increase with PEG size, as larger PEG sizes display a stronger departure
from linearity, which is amplified at lower temperature (
Extended Data Fig.1d
). Similarly, as can be
observed in Supplementary Figure 1, Equation 2 fails to model the osmometry curves of BSA as a
function of concentration, which exhibit a much stronger departure from linearity than PEG.
Supplementary Figure 1
: Osmotic potential of indicated solutions fitted to various models.
We thus considered an alternative way to model our curves, and, following the work of
Fullerton and colleagues
4,5
, fitted our data to equation (3), with
퐼퐼
푒푒푒푒푒푒
푠푠
the
effective
interaction term
(Equation 3 corresponds to Equation 1 in the main text
). Note that
a priori
,
퐼퐼
푒푒푒푒푒푒
푠푠
and
퐴퐴
may be
function
s
that vary with solute identity and temperature
.
−훹훹
휋휋
(
퐶퐶
,
푇푇
) =
퐴퐴
(
푇푇
)
×
퐶퐶
1
−퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
) ×
퐶퐶
(3)
As can be observed in Supplementary Figure 1, Equation (3) fits well the osmometry curves of
PEG 35kDa, and is markedly better at fitting the osmotic potential of BSA, which exhibits a much
stronger departure from linearity than PEG.
Since
PEG
measurements were performed at multiple temperatures, we thought to constrain
the fits to avoid fitting two parameters (
퐴퐴
(
푇푇
)
and
퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
)
) with only one curve. Indeed,
an interesting
limit case of equation (3) is when
퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
)
×
퐶퐶 ≪
1
. In these conditions,
퐴퐴
tends to
푖푖
푅푅푇푇
(
that is
the
van
't Hoff
equation,
equation 1
), thus, we hypothesized that A should be linear with temperature, and
thus fitted simultaneously all curves from different temperatures to equation 4
.
−훹훹
휋휋
(
퐶퐶
,
푇푇
)
≈
퐴퐴
′
푇푇
×
퐶퐶
1
−퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
)
×
퐶퐶
(4)
Supplementary Figure 2: Variation of the value of virial coefficient as a function of temperature
and PEG size. This analysis was performed by considering a model of order two or three (that is
one or two virial coefficients).
In this case,
퐴퐴
′
is a constant that is identical for all the datasets
from a given macromolecule
,
meaning that for three datasets at
three
different temperatures, only four parameters are considered
,
rather than six. This provided similar results to fitting each curve to equation 3
, albeit with better
goodness of fits (for our PEG data set of 8 molecular weight
s and 3 temperatures
, so 24 datapoints,
Spearman correlation
of
0.9948
and
p<0.0001
between the values of
퐼퐼
푒푒푒푒푒푒
푠푠
obtained by fitting each
curve independently or all curves simultaneously
).
As can be observed in Supplementary Figure 3, fitting our dataset to Equation (4) provided
much more consistent results as a function of PEG size and temperature compared with Equation (2).
We found that
퐼퐼
푒푒푒푒푒푒
푠푠
scales quasi
-linearly with PEG size, and the slope of this curve is higher at lower
temperature. Note also that
퐴퐴
′
scales quasi
-linearly with PEG size, which is expected: A’ is related to
푖푖
, the
van’t Hoff
factor, which is expected to increase with PEG size. Thus, throughout this paper, we
systematically fit our data using equation (4) and used
퐼퐼
푒푒푒푒푒푒
푠푠
as an empirical measure of the impact of
the solute upon the solvent.
Note that equation 4 can be rearranged as equation 5. This highlights the two contributions
to the osmotic potential:
퐴퐴
′
푇푇퐶퐶
models the effect of an ideal solute on the solvent, whereas
1
1− 퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
)
퐶퐶
models the extent of the unfavourable interactions at the solute:solvent interface.
−훹훹
휋휋
(
퐶퐶
,
푇푇
)
≈퐴퐴
′
푇푇퐶퐶
×
1
1
−퐼퐼
푒푒푒푒푒푒
푠푠
(
푇푇
)
퐶퐶
(5)
Supplementary Figure 3
: Variation of the value of
퐼퐼
푒푒푒푒푒푒
푠푠
as a function of temperature and PEG size.
This analysis was performed by considering
equation 4.
Chapter II/ Other frameworks for modelling polymer phase behaviour and
their
application
to
biomolecular condensation in cells.
Most theoretical frameworks for phase separation consider entropic and enthalpic effects from the
standpoint of the polymer. For example, Flory
-Huggins solution theory (FHT) was developed to
describe the thermodynamics of polymer
-solvent mixtures, expressing the free energy of mixing in
terms of solvent and polymer volume fractions, temperature and a single interaction parameter
χ
to
describe the strength of solvent-polymer interactions compared with average solvent-solvent and
polymer
-polymer interactions
6
. This lattice model considers the entropic and enthalpic drivers of
polymer behaviour, with the value of
χ
predicting whether the mixture is homogeneous or forms two
coexisting phases, and was subsequently extended to mixtures of two or more polymers by Scott and
Tompa
7,8
. FHT has been successfully used to model the phase behaviour of, and fit data from,
macromolecular solutions
in vitro,
where condensation is examined as a function of the concentration
of the specific protein(s) under investigation (e.g.
9
), but makes several critical assumptions that limit
its direct applicability to proteins and their interactions within cells
:
•
FHT assumes the polymer is made from identical (non
-polar) monomers. This is not true for any
proteins in cells, as the cytosol is composed from a mixture of thousands of different proteins,
whose relative abundance varies by several orders of magnitude.
•
Under the FHT mean field approximation, all polymer segments interact equally. This is not true
for any natural polypeptides which are composed from amino acid residues with differing
chemical properties (charged vs uncharged, hydrophilic vs hydrophobic, b
ulky vs small, aromatic
vs aliphatic etc), and whose activity is further modulated by site
-specific post
-translational
modifications such as phosphorylation.
•
FHT assumes ideal polymer chain behaviour, and does not take into account chain connectivity
-
interactions between different polymer regions or any resulting topological constraints. Such
interactions occur in almost all cellular proteins, leading to seco
ndary and tertiary structures that
are quite specific to the primary amino acid sequence, with most soluble proteins in the cytoplasm
adopting compact globular structures.
•
χ
is used to describe all interactions in the system, and so cannot adequately account for specific
interactions critical to the behaviour of specific proteins in biological systems (e.g. electrostatic
interactions, hydrophobic interactions and hydrogen bonding). Indeed, the physics of
hydrophobic surface hydration itself are not sufficiently well understood that the free energy
change can be predicted reliably
10
.
•
FHT considers a homogenous solvent, whereas the cytosol is a highly complex, heterogeneous and
crowded environment, in which the activity of hydration water has been experimentally
distinguished from bulk solvent by multiple independent methods. These heterogeneities and
spatial constraints are not taken into consideration by FHT
10–
12
.
•
FHT assumes that polymers and solvent molecules can move freely and do not impose constraints
on each other’s mobility. This is not the case for water molecules hydrating proteins and other
macromolecules, which have reduced translational and rotational en
tropy compared with bulk
solvent. Moreover, the free energy of solvent
-sidechain interactions differs enormously between
hydrophobic and polar/charged amino acid residues, and also varies with context
10
.
Overall
then
, while FHT can accurately describe the phase behaviour of dilute binary polymer blends,
it does not describe the unique behaviours of water as a solvent, nor does it capture the behaviour of
complex macromolecular mixtures found in biological systems with
out substantial modifications and
extensions that try to account for specific interactions and factors such as chain connectivity and
molecular crowding. The review by Zaslavsky and Uversky
13
provides a more detailed perspective on
these themes, highlighting that reversible condensation of natively folded proteins does not occur in
any solvent besides water and its importance to this process.
More recent attempts to model the thermodynamics of protein condensation in cells have explicitly
considered changes in free energy of the solvent in a more granular fashion than FHT
14
, but in general,
models of cellular phase separation focus on the enthalpic driving force generated by weak,
multivalent interactions between macromolecules. Solvent entropy rarely receives the same level of
consideration even though water is the most abu
ndant molecular species in any biological system and
thus a major factor determining its thermodynamical equilibria. To our knowledge, the multitude of
different theoretical frameworks that aim to describe liquid-liquid phase separation of proteins do not
predict the non
-linear relationship between BSA concentration and osmotic potential, for example,
nor the interaction with temperature we observe across the physiological range. By focussing on the
solvent and employing the Fullerton empirical
model that explicitly considers the interaction between
polymers and water
4,5
, as described above, our work provides a framework for understanding the
effects of temperature on the behaviour of concentrated polymer solutions and the impact of
manipulations of solvent thermodynamics
, e.g.,
heavy water. Given the concentrated, colloidal
intracellular environment,
we hope that this approach will aid our understanding of the physiological
drivers and functions of phase separation in cells
.
Supplementary References
1.
Scatchard, G. Physical Chemistry of Protein Solutions. I. Derivation of the Equations for the
Osmotic Pressure.
Journal of the American Chemical Society
68
, 2315–
2319 (1946).
2.
Vink, H. Precision measurements of osmotic pressure in concentrated polymer solutions.
European Polymer Journal
7
, 1411–
1419 (1971).
3.
Mitchison, T. J. Colloid osmotic parameterization and measurement of subcellular crowding.
Molecular Biology of the Cell
30
, 173–
180 (2019).
4.
Zimmerman, R. J., Kanal, K. M., Sanders, J., Cameron, I. L. & Fullerton, G. D. Osmotic pressure
method to measure salt induced folding/unfolding of bovine serum albumin.
J Biochem
Biophys Methods
30
, 113–
131 (1995).
5.
Cameron, I. L., Kanal, K. M., Keener, C. R. & Fullerton, G. D. A mechanistic view of the non
-
ideal osmotic and motional behavior of intracellular water.
Cell Biol Int
21
, 99–
113 (1997).
6.
Flory, P. J. Thermodynamics of High Polymer Solutions.
J Chem Phys
10
, 51–
61 (1942).
7.
Tompa, H. Phase relationships in polymer solutions.
Transactions of the Faraday Society
45
,
1142–
1152 (1949).
8.
Scott, R. L. The Thermodynamics of High Polymer Solutions. V. Phase Equilibria in the Ternary
System: Polymer 1
—
Polymer 2
—
Solvent.
J Chem Phys
17
, 279–
284 (1949).
9.
Nott, T. J.
et al.
Phase Transition of a Disordered Nuage Protein Generates Environmentally
Responsive Membraneless Organelles.
Mol Cell
57
, 936–
947 (2015).
10.
Rego, N. B. & Patel, A. J. Understanding Hydrophobic Effects: Insights from Water Density
Fluctuations.
https://doi.org/10.1146/annurev
-conmatphys
-040220-045516
13
, 303–
324
(2022).
11.
Persson, E. & Halle, B. Cell water dynamics on multiple time scales.
Proc Natl Acad Sci U S A
105
, 6266–
6271 (2008).
12.
Ebbinghaus, S.
et al.
An extended dynamical hydration shell around proteins.
Proc Natl Acad
Sci U S A
104
, 20749–
20752 (2007).
13.
Zaslavsky, B. Y. & Uversky, V. N. In Aqua Veritas: The Indispensable yet Mostly Ignored Role of
Water in Phase Separation and Membrane-less Organelles.
Biochemistry
57
, 2437–
2451
(2018).
14.
Bartolucci, G., Michaels, T. C. T. & Weber, C. A. The interplay between molecular assembly
and phase separation.
bioRxiv
2023.04.18.537072 (2023) doi:10.1101/2023.04.18.537072.