SLIDE 1 Funnel Gas Model for Protein Many-Body Systems under the Crowded Environment
Macoto Kikuchi1, Yoshikatsu Tada1, Nobu C. Shirai2
1.Cybermedia Center, Osaka Univ. 2.Center for Information Technologies and Networks, Mie Univ.
SLIDE 2 Goal of the talk
1
Introduce a simple lattice-gas type model to describe the system consisting of proteins and crowding molecules.
2
Apply the effective free-energy of the model to discuss the crowding effect on the folding of small globular proteins
SLIDE 3
Part 1
Construction of the Funnel Gas Model
SLIDE 4 Motivation
Background Foldable proteins have funnel-like energy landscape
Energy landscape theory or funnel picture (e.g. Wolyness)
Interior of the living cells is crowded by biological molecules
Molecular crowding effect (e.g. Minton)
SLIDE 5
Purpose To simulate the protein many-body system with crowding 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.
SLIDE 6
Setup
Proteins A protein has internal states, for which the funnel-like free-energy landscape is taken into account: FUNNEL GAS! The native state occupies a single site A few denatured states that occupy one or more sites (expanded) Crowding molecules A crowding molecule occupy a single site Interact only through on-site excluded volume effect (no other interactions considered)
SLIDE 7
Crowder Native Denatured Denatured
Concept of Funnel Gas Model
SLIDE 8 A Little bit of theory
Protein fi : Free energy of i-th state vi : Volume (number of the occupation sites)
ni : Number of proteins at i-th state
∑
i ni = N : Total number of proteins
Protein-protein interactions can also be taken into account.
SLIDE 9 System Fixed number of proteins + Grand-canonical ensemble of crowders Partition Function Ξ = Tr [ M ∑
m=0
(M m ) eβµme− ∑
i βfini
] β: Inverse temperature µ: Chemical potential of the crowders M: Number of the free sites on which the crowders can be placed
SLIDE 10 Using the binomial theorem, Ξ is rewritten as Ξ = eαVTr [ e− ∑
i(βfi+αvi)ni
] , where α = log(1 + eβµ) The crowders are traced out.
SLIDE 11 Effective Free Energy for a Protein βf eff
i
= βfi + α vi Volume dependent modification
also depends on the density of crowders through α
Effective Partition Function Z eff = Tr [ e−β ∑
i f eff i
ni
] Equivalent to the canonical ensemble consisting only of proteins with the effective free-energy f eff
i
SLIDE 12 We can simulate the protein many-body systems under the crowding environment without explicitly considering the crowding molecules.
Computational effort is largely reduced.
SLIDE 13 Limitation of the model
Crowding molecules should not be too small
The depletion force (Asakura and Osawa) is not taken 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 the size of the crowding molecules are comparable to that of the native state of the protein
SLIDE 14
Part 2
Molecular Crowding Effects on Protein Folding
SLIDE 15 Motivation
Effective free-energy derived for the funnel gas model βf eff
i
= βfi + α vi By identifying vi as the volume of i-th conformation, we can use f eff
i
to obtain an approximate free-energy landscape of the protein for any model
SLIDE 16 Application
Study the effect of molecular crowding on the folding of small globular proteins using f eff
i .
Model Go model on SC lattice for protein
This lattice is different from the one used in the funnel gas model To calculate f eff
i
, the protein is considered as a sphere of the radius Rg (radius of gyration).
Crowders are treated by the funnel gas model
Size of the crowders is taken as the same as that
- f the native state of the protein
SLIDE 17
Native conformations of toy proteins
Up-Down β Burrel Four-Helix Bundle
SLIDE 18
Free-energy landscapes for proteins without crowders are obtained using the Multi-Self-Overlap-Ensemble Monte Carlo method Chikenji, Kikuchi and Iba, PRL 83 (1999) 1886.
SLIDE 19
Result: Up-Down β burrel
SLIDE 20 4 3 5 3 2 5 2 1 5 1 5 E 2 3 4 5 6 7 Rg
α = 0.0
43.94 41.94 39.94 37.94 35.94 33.94
Free energy landscape at Tf (no crowder)
SLIDE 21 4 3 5 3 2 5 2 1 5 1 5 E 2 3 4 5 6 7 Rg
α = 0.32
44.94 42.94 40.94 38.94 36.94 34.94 21.66
crowder volume fraction φ ≃ 0.2
SLIDE 22 4 3 5 3 2 5 2 1 5 1 5 E 2 3 4 5 6 7 Rg
α = 0.78
44.94 42.94 40.94 38.94 36.94 34.94 2.02
crowder volume fraction φ ≃ 0.4
SLIDE 23 4 5 4 3 5 3 2 5 2 1 5 1 5 E 45 44 43 42 41 40 39 f
α = 0.0 α = 0.32 α = 0.78
Energy vs. Free Energy at Tf
SLIDE 24
Result: Four-Helix Bundle
SLIDE 25 1 8 6 4 2 E 4 6 8 10 Rg
α = 0.0
104.15 101.15 98.15 95.15 92.15 89.15 86.15
Free energy landscape at Tf (no crowder)
SLIDE 26 1 8 6 4 2 E 4 6 8 10 Rg
α = 0.32
106.15 103.15 100.15 97.15 94.15 91.15 88.15 41.83
crowder volume fraction φ ≃ 0.2
SLIDE 27 1 8 6 4 2 E 4 6 8 10 Rg
α = 0.78
106.1 103.1 100.1 97.1 94.1 91.1 88.1 22.0
crowder volume fraction φ ≃ 0.4
SLIDE 28 1 2 1 8 6 4 2 E 110 105 100 95 90 85 f
α = 0.0 α = 0.32 α = 0.78
Energy vs. Free Energy at Tf
SLIDE 29 The native state is stabilized by the crowding
- effect. ← Expanded conformations of the
denatured 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.
SLIDE 30 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 the effective free energy of the protein as long as
- nly the excluded volume interaction is taken
into account. We applied the effective free energy to folding
- f small globular proteins and showed that the
native states are stabilized by destabilization of the expanded denatured state due to the molecular crowding effect.
