Funnel Gas Model for Protein
Many-Body Systems under theCrowded Environment
Macoto Kikuchi1 , Yoshikatsu Tada1 ,Nobu C. Shirai2
1.Cybermedia Center, Osaka Univ.
2.Center for Information Technologies and Networks, Mie Univ.

Goal of the talk1

2

Introduce a simple lattice-gas type model todescribe the system consisting of proteins andcrowding molecules.
Apply the eﬀective free-energy of the model todiscuss the crowding eﬀect on the folding of
small globular proteinsPart 1

Construction of the FunnelGas ModelMotivation
Background
Foldable proteins have funnel-like energy
landscapeEnergy landscape theory or funnel picture (e.g.
Wolyness)

Interior of the living cells is crowded bybiological molecules
Molecular crowding eﬀect (e.g. Minton)Purpose
To simulate the protein many-body system withcrowding molecules, we need an appropriately
coarse-grained model.
We develop a Lattice-gas based model
A prototype model has already been introduced in
Shirai and Kikuchi, J.Chem.Phys. 144 (2016)
055101.

SetupProteins
A protein has internal states, for which thefunnel-like free-energy landscape is taken into
account: FUNNEL GAS!
The native state occupies a single site
A few denatured states that occupy one ormore sites (expanded)
Crowding molecules
A crowding molecule occupy a single site
Interact only through on-site excluded volume
eﬀect (no other interactions considered)Denatured
NativeDenatured
Crowder
Concept of Funnel Gas ModelA Little bit of theory
Proteinfi : Free energy of i-th statevi : Volume (number of the occupation sites)of i-th stateni : Numberof proteins at i-th state∑i ni = N : Total number of proteinsProtein-protein interactions can also be taken intoaccount.

SystemFixed number of proteins+ Grand-canonical ensemble of crowders
Partition Function][M ( )∑ M∑
Ξ = Tre βµm e − i βfi nim
m=0
β: Inverse temperatureµ: Chemical potential of the crowders
M: Number of the free sites on which thecrowders can be placed

Using the binomial theorem, Ξ is rewritten as[ ∑]αV− i (βfi +αvi )niΞ = e Tr e,
whereα = log(1 + e βµ )The crowders are traced out.

Eﬀective Free Energy for a Protein
βfi eﬀ = βfi + α vi
Volume dependent modification
also depends on the density of crowders through αEﬀective Partition Function[]∑eﬀ−β i fi eﬀ ni
Z = Tr e
Equivalent to the canonical ensemble
consisting only of proteins with the eﬀective
free-energy fi eﬀ

We can simulate the protein many-bodysystems under the crowding environment
without explicitly considering the crowding
molecules.
Computational eﬀort is largely reduced.

Limitation of the model
Crowding molecules should not be too small
The depletion force (Asakura and Osawa) is nottaken into account.

Crowding molecules should not be too large
The volume of denatured states does no eﬀect

The model is suitable for the case that thesize of the crowding molecules are comparable
to that of the native state of the protein

Part 2

Molecular Crowding
Eﬀects on Protein Folding

Motivation
Eﬀective free-energy derived for the funnel gas
model
βfi eﬀ = βfi + α vi
By identifying vi as the volume of i-th conformation,
we can use fi eﬀ to obtain an approximate free-energy
landscape of the protein for any model

Application
Study the eﬀect of molecular crowding on thefolding of small globular proteins using fi eﬀ .
ModelGo model on SC lattice for protein
This lattice is diﬀerent from the one used in thefunnel gas model
To calculate fi eﬀ , the protein is considered as asphere of the radius Rg (radius of gyration).

Crowders are treated by the funnel gas modelof closed-packed lattice.
Size of the crowders is taken as the same as thatof the native state of the protein

Native conformations of toy proteins

Up-Down β Burrel

Four-Helix Bundle

Free-energy landscapes for proteins withoutcrowders are obtained using theMulti-Self-Overlap-Ensemble Monte Carlo methodChikenji, Kikuchi and Iba, PRL 83 (1999) 1886.

Result: Up-Down β burrelRg

α = 0.076543240 35 30 25 20 15 10 5 0

33.94
35.94
37.94
39.94
41.94
43.94

EFree energy landscape at Tf (no crowder)Rg

α = 0.32
76543240 35 30 25 20 15 10 5 0

21.66
34.94
36.94
38.94
40.94
42.94
44.94

Ecrowder volume fraction ϕ ≃ 0.2

Rgα = 0.7876543240 35 30 25 20 15 10 5 02.02
34.94
36.94
38.94
40.94
42.94
44.94

Ecrowder volume fraction ϕ ≃ 0.4

f39 α = 0.040 αα == 0.32
0.78414243444545 40 35 30 25 20 15 10 5 0
EEnergy vs. Free Energy at Tf

Result: Four-Helix Bundle

Rg

α = 0.0

86.15
89.15
92.15
95.15
98.15
101.15104.1510864

100 80 60 40 200E
Free energy landscape at Tf (no crowder)Rg

α = 0.32
10864

100 80 60 40 20041.83
88.15
91.15
94.1597.15
100.15103.15106.15E
crowder volume fraction ϕ ≃ 0.2

Rg

α = 0.78
10864

100 80 60 40 20022.0
88.1
91.1
94.1
97.1
100.1
103.1
106.1

E
crowder volume fraction ϕ ≃ 0.4

8590

α = 0.0
α = 0.32
α = 0.78

f95100105110
120 100

80

60

4020E
Energy vs. Free Energy at Tf

0The native state is stabilized by the crowding
eﬀect. ← Expanded conformations of thedenatured state becomes unstable
Although this eﬀect has long been expected
intuitively or from some simulation results (e.g.
Cheung et al. (2005)), the present formulation
clearly explains the origin of the eﬀect.

Summary
We formulated the funnel gas model for
simulating protein many-body systems under
the crowded environment.
The eﬀect of the crowder is absorbed in theeﬀective free energy of the protein as long as
only the excluded volume interaction is taken
into account.
We applied the eﬀective free energy to foldingof small globular proteins and showed that thenative states are stabilized by destabilization of
the expanded denatured state due to themolecular crowding eﬀect.