theoretical perspectives on biomolecular...
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JBB 2026H • Lecture 7 (2nd half) Hue Sun Chan
Department of Biochemistry October 26, 2018
University of Toronto
Theoretical Perspectives on Biomolecular Condensates
Image from: Banani, Lee, Hyman & Rosen, Nat Rev Mol Cell Biol 2017
Intracellular Biomolecular Condensates or “Membraneless Organelles” are organizers of biochemistry
Theory and
Computational
Simulation of
Multiple-Chain Systems
• Rubber elasticity: An application of the Gaussian chain theory
Figure Credit: Pomès Group, The Hospital
for Sick Children & University of Toronto
where
Note: R = kB (Avogadro #)
Figure from: Rauscher & Pomès, “Structural disorder and protein elasticity”. In: Fuzziness: Structural
Disorder in Protein Complexes. Edited by Fuxreiter & Tompa. Springer (2012).
Mean-field Flory-Huggins Theory addresses sequence composition of intrinsically
disordered proteins (IDPs) such as overall
hydrophobicity but not other encoded information
such as hydrophobic & charge patterns, etc. in the
sequence.
• “Mean-field” refers to approximations in
which some fluctuations/variations in IDP
concentrations and/or sequence-dependent
interactions, etc., are averaged/smeared over.
= total solvent volume
B
Flory-Huggins uses a conceptual
(non-explicit) lattice argument
z = lattice coordination number
# of ways of placing the first bead
of the first chain = V
# of ways of placing the second
bead = (z ‒ 1) (probability that
the site is not occupied). This
probability is given by (V ‒ 1)/V
in the mean-field approximation
Therefore, the # of ways of
placing the entire first chain in
this approximation is
[(z ‒1)(V‒1)/V][(z ‒1)(V‒2)/V]…
… [(z ‒1) (V‒m+1)]
= [V !/(V‒m)!][(z ‒1)/V]m-1
• Then repeat the argument for
the other N ‒ 1 chains. The N chains are identical
When b → solvent (s), mb = 1,
a → p (polymer/protein),
ϕs = 1 ‒ ϕp ,
the contact free energy per unit
volume in the mixed state
E/V=z [εpp ϕp2/2 +
εss (1 ‒ ϕp )2/2 +
εps ϕp (1 ‒ ϕp)],
whereas the contact free energy per
unit volume in the unmixed state
=z [εpp ϕp/2 + εss (1 ‒ ϕp) /2].
Hence the contact free energy per
unit volume = (former) – (latter) =
Δ“H”mix/V = kBT χ ϕp(1 ‒ ϕp)
E/V=z ( )
B
Lin, Forman-Kay & Chan, Biochemistry 57:2499-2508 (2018)
Depending on the IDP sequence, phase separation can be induced
by an increase or a decrease in temperature
= χ-1(T)
Image Credit:
Muiznieks & Keeley 2013
Study.com
LCST example:
tropoelastin
Dill, Hutchinson & Alonso 1989
Binodal and Spinodal Phase Separations
Single-phase globally unstable ⤍ Spinodal (Global) Phase Separation, Spinodal Decomposition
Binodal (Local) Phase Separation, Droplets
Spinodal region lies inside
the coexistence region
Theoretical Formulations Available at Various Levels of Structural and Energetic Details
Flory-Huggins Overbeek-Voorn RPA Advanced Theory Explicit-Chain
IDP phase separation is sequence dependent : not only depends on composition
Random Phase Approximation (RPA) is an analytical theory of sequence-dependent
polyampholyte (polymer chains with both + and ‒ charges) phase separation
Lin, Forman-Kay & Chan, Biochemistry 57:2499-2508 (2018)
RNA helicase Ddx4: Wildtype forms
condensates in cell or in vitro, but a
charge-scrambled mutant does not
Sequence-dependent RPA theory
captures this trend:
Nott et al., Forman-Kay & Baldwin, Mol Cell (2015)
Protein volume fraction ϕm
binodal
spinodal Tem
pera
ture
T*
(th
eo
retica
l u
nits)
Lin, Forman-Kay & Chan, Phys Rev Lett (2016); Lin, Song, Forman-Kay & Chan, J Mol Liquids (2017)
Spinodal decomposition in Ddx4N1 drying experiment
Sawle & Ghosh, J Chem Phys (2015)
Das & Pappu, PNAS (2013)
Sequence charge pattern parameters align well with
RPA-predicted phase separation propensity
Y.-H. Lin & H.S. Chan, Biophys J (2017)
beyond simple “blockiness”?
same number of + and ‒ charges
Das & Pappu, PNAS , 2013
Single-chain conformational compactness and multiple-chain
phase separation are favored by similar block-like charge
patterns that promote sequence-nonlocal attraction
Y.-H. Lin & H.S. Chan, Biophys J (2017)
Phase Separation as a Mechanism for Homeostasis
Compositions of the phase-separated states are less varied than the underlying bulk concentrations
Lin, Forman-Kay & Chan, Biochemistry 57:2499-2508 (2018)
FIB1
NPM1
Image credit: Marina Feric & Cliff Brangwynne,
Princeton University.
From: New J Phys Focus on “Phase Transitions in Cells:
From Metastable Droplets to Cytoplasmic
Assemblies”
How do different IDPs find one another to from the many separate intracellular compartments and subcompartments? Why don’t they all condense together into a large gemisch? A multivalent, stochastic, “fuzzy” mode of molecular recognition?
nucleoli
The Binary Phase Diagram (Pattern of Coexistence) for a Pair of Polyampholytes
Varies Significantly with the Charge Patterns Along their Sequences
Lin, Brady, Forman-Kay & Chan, New J Phys (2017)
The two IDP components co-mix when their charge patterns are similar (as measured by κ or SCD); de-mix when their charge patterns are dissimilar.
SCD1‒ SCD2
= ‒1.01 SCD1‒ SCD2
= 8.62
SCD1‒ SCD2
= 15.58
Assessing analytical theories and rationalizing experiments by explicit-chain simulations
Das, Amin, Lin & Chan, q-bio-arXiv:1808.10023 (2018)
See also: Das, Eisen, Lin & Chan, J Phys Chem B 2018, Dignon, Zheng, Kim, Best & Mittal,
PLoS Comput Biol 2018; Silmore, Howard & Panagiotopoulos, Mol Phys 2017
Below Tcr : Above Tcr : Marginal? :
500 Cα-chains
Current sequence charge pattern parameters are predictive of simulated
phase behaviors to a certain degree for some polyampholyte sequences
• Fully charged sequences
with more blocky charge
patterns have higher
UCSTs (upper critical
solution temperatures),
i.e., higher propensities
to phase separate.
Simulated results from: Das, Amin, Lin & Chan, q-bio-arXiv:1808.10023 (2018)
Dignon et al., PNAS (2018) doi.org/10.1073/pnas.1804177115 Dignon et al., PLoS Comput Biol 14:e1005941 (2018)
Applications of the residue-based model
∎ Correlation between single-chain collapse and multiple-chain phase separation propensities (cf. Lin & Chan)
∎ Lower phase separation propensities of FUS variants with phosphomimetic mutations (as in experiment).
Results and figures from: Harmon, Holehouse & Pappu, New J Phys 20: 045002 (2018); see also Feric et al., Cell 165:1686 (2016) and Harmon et al., eLife 6:e30294 (2017).
Lattice Models in which each lattice site (bead)
represents a multiple-residue group/domain,
beads are connected by disordered linkers
i.e., poly-(proline-rich module)
Example (∼experimental system studied in the
Mike Rosen lab):
More complex scenarios with liquid-liquid, liquid-gel, and gel-gel phase separations
Illustrative example using
the extended FH theory of
Semenov & Rubinstein
[Macromolecules 31:1373
(1998)], with N = 100,
# of stickers/chain = f = 5.
[cf. lattice model of
Harmon, Holehouse, Rosen
& Pappu, eLife 6:e30294
(2017)]
Figure from: Lin, Forman-Kay & Chan, Biochemistry 57:2499-2508 (2018)