生体高分子の粗視化分子シミュ
レーションで何を見る？
理研QBiC 分子機能シミュレーション研究チーム
安藤 格士
バイオスーパーコンピューティング研究会(BSCRC)
ウィンタースクール 2015 @休暇村伊良湖 1/30/2015

自己紹介•大学では分子生物学、生化学等、を学びました。修士課程（Ｙ研究室）までは、大腸菌での遺伝子操作、タンパク質の発現、精製を行い分光学的手法（ＣＤ，ＮＭＲ）でタンパク質の安定性を調べていました。私のバックグラウンドは生化学、分子生物学です。

•

博士課程（Ｙ研究室）では、ブラウン動力学法を用いたタンパク質の折り畳みシミュレーションを行っていました。

•アメリカに留学（Skolnick Lab.）し、細胞内環境下でのシミュレーション等を行いました。

•

日本に帰国後、理研（Ｓ研究室）で生体高分子のシミュレーションに関わる仕事をしています。

•物理化学・理論、そして数学には弱いですが、とても憧れがあります。 曖昧模糊
で、結果の記述が多い、複雑な生物学の分野で、何か単純な物理化学的な原理を見つけ出し、予測可能な理論的なものを見つけ出せないかなー、と考えています。

今回の発表• 今回は非常にリラックスした環境で発表ができるということで、あまり注目されませんでしたが、私的には気に入っている研究についてお話をさせていただきます。そして、最後に分子シミュレーション、特に粗視化分子シミュレーションに対する個人的な考えを少しだけ（２スライド）述べさせていただきます。

Force-field for two-points lipid model• Bonding term

In the simulations, kbond of 1000kBT/r02 and r0 of13 Å were used.

HHH

TT

T

T

TTH

12
 k bond r  r0 2HVbond• Non-bonding termV  V

atr
inter-lipidT-T pairs

repall inter-lipidpairs  L  r  rc  Vatr    L cos 2 2wc
0
Vrep 1
2k rep r  rc 2

if r  rc
if rc  r  rc  wc

HV inter if r  rc  wc
if r  rc

In the simulations, εL of 1.9kBT, wc of 19.5 Å, krepof 10kBT were used.

Equilibrium distances (Å) between
inter-monomer particles, rc

Particle #HTH

11.70

12.35

T

12.35

13.00

ブラウン動力学(BD)• Effects of solvent on solute molecules are implicitlyconsidered. (Coarse-graining!!)It makes possible to examine long-time dynamics. (Inprinciple, water molecules should be explicitly presentin MD simulation systems.)• Stochastic simulation based on the Langevin equation.

• Hydrodynamic interactions (HI) can be taken into account.

BD algorithm
粒子間の流体力学的相互作用を考慮しない場合• BD algorithm considering HIDFr t  t   r t  t  2t Bzk BT

拡散行列Ｄは対角行列with D  BBTr is a 3N vector representing the spatial coordinates of
all the particles in the simulation box. D is the diffusion
tensor with size of 3N x 3N. F is a 3N vector of force. z isa 3N vector with zero mean and variance of 1.

• Rotne-Prager-Yamakawa tensor
k T B I 6a2a 2  1
 k BT ˆˆˆˆ
Dij   I  rij rij  2  I  rijrij 8r
rij  3
ij  k BT 1  9 rij I  3 rij rˆ rˆ  6a  32 a  32 a ij ij 



kBTD
6 a
ストークス - アインシュタインの式
BD法では流体力学的相互作用を自在にON/OFFできます！

i  j,i  j and rij  2a,i  j and rij  2a.

• MDではできない。
• 生体高分子のＭＤを行っている研究者のほとんどは、流体力学的相互作用に注目していない。

What are hydrodynamic interactions?
Each particle’s force changes the solvent flow, and this in turnaffects forces on other particles through the frictional forces
affecting them.

FSystems for membrane assembly• 1000 coarse-grained lipid molecules were placed randomly in a 28 × 28
× 28 nm3 periodic box without significant overlaps between particles.
• Ten independent initial configurations were generated.
• Simulation temperature was set to 298 K.
• Stokes radii of particles, a, were set to 6.5 Å.
• Time step was 2.24 ps, which corresponds to 0.5 × 10−3 a2/D0 with D0 =kBT/6πηa.
• HIs were considered by two different ways:– BD simulation with full HIs. That is, HIs within each lipid molecule as well as
between lipid molecules (With inter-HIs)– BD simulation with only intramolecular HIs, where intermolecular HIs areneglected (Without inter-HIs)

二つの条件の違いは、分子間の流体相互作用がある（with inter-HIs）,なし（without inter-HI）のみ。 → ２つの結果の差から分子間流体力学の影響を見ることができる。

Representative simulation results
With inter-HIsWithout inter-HIs

3.5 μs BD simulations

HI accelerate membrane self-assemblyTime evolution of nematic order parameter S

To measure order of lipid membrane, we evaluated the nematic orderparameter S of the system which is given by the largest eigenvalue of anorder parameter defined by 3 × 3 matrix
1Q  3 cos   cos      with α,   x, y, z
2Here, θα is the angle between a lipid molecule axis and α axis, δαβ is theKronecker delta function, and represents the average over all molecules inthe system. S equals 1 for perfectly aligned molecules and 0 for a random
configuration.

Overall kinetics of membrane self-assembly

f is the fraction of systems that reach S > 0.7 within given time periods.

k(with HI)/k(without HI) = 6.4

分子間ＨＩは、膜形成速度を約６倍加速させている。

HI decelerate initial monomermonomer association
Growth of average lipid-cluster size <Nlipid>
up to 0.1 μs

The ratios of the slopes of the growth curves in
the presence and absence of inter-HI for various<Nlipid>• <Nlipid> = 1 では, g(with inter-HI)/g(without inter-HI) = 0.50.
=> 分子間ＨＩは、モノマー・モノマーの結合速度を遅くしている。
• しかし、 <Nlipid> が 5以上になると、分子間ＨＩを考慮した方が、脂質クラスターの融合速度が速い。

How can we explain this size
dependent HI effects?
In a diffusion controlled kinetic, the diffusive encounter
rate, kD, can be described by-1é ¥ exp éëW ( r ) kBT ùû ùkD = 4p ê òdrú2sD (r) rêëúûwhere W(r) is the potential mean force between thereactants at center-to-center distance r, σ is the encounter
distance, and D(r) is the distance-dependent relativetranslational diffusion coefficient. When each reactant isrepresented by a sphere, D(r) at diluted condition can bewritten by

a1a2r
Diffusion
Reaction!

D ( r ) = D1 + D2 - 2r̂12 × D12 × r̂12Here, D12 is diffusion tensor between particles 1 and 2, rr̂12
12
is the unit vector between reactants, and Di is thediffusion coefficient of particle i given byDi k BT
6πηaiσHow can we explain this size
dependent HI effects?
Let us assume W(r) = 0, the reactants have a same radius a0, and σ is the sum of a0. In theabsence of HI, that is “free-draining (FD) limit”, the relative diffusion coefficient is a constant andsum of the diffusion coefficients of the reactants, that is D(r) = 2D0. Therefore, the rate constantin FD limit, kDFD, is given byThis is the Smoluchowski expression.
In the presence of HI at Oseen tensor level, D(r) is given by

D ( r ) = 2D0 Then, the encounter rate , kDOseen, is expressed as

kDOseen =kBT2phr

D(r)/D0kDFD =16p D0 a0

12p D0 a0ln 4When the reactants are represented by single particles having the same radius,

kDOseen» 0.54FDkD

HI reduce the relative
diffusion coefficient

r/a0This means HI reduce the diffusive encounter rate by 46% (Deutch and Felderhof, 1976).

How can we explain this size
dependent HI effects?
Next, we consider the diffusive encounter rate of objects
consisting of N beads, like polymers. The diffusion
coefficient of a random polymer connecting N identicalbeads in the presence of HI shows the following scaling
property:0   1DOseen  D0 N Here, DOseen is the diffusion coefficient of the polymer.
The hydrodynamic radius of the polymer a is related to its
DOseen byk BTa
 a0 N Oseen
6DThus, the radius of polymer has Nν scaling. Therefore, thediffusive encounter rate in the presence of HI is given by
12D Oseena 2k BTOseenkD ln 4
 ln 4
The rate does not show any N dependence for thereactant pair consisting of the same number of beads.

aaDiffusion
Reaction!

σ = 2aHow can we explain this size
dependent HI effects?
In the FD limit, the diffusion coefficient DFD is given by Rouse theory, which isD FD  D0 N 1
Because HI only affects kinetics and not thermodynamics, polymers have the sameradius in the presence and absence of HI, which scales with Nν. Therefore, we obtainthe following scaling properties for the encounter rate in the FD limit:k DFD  16D FD a 8k BT  1
N3

which shows N-dependence.
The ratio of the diffusive encounter rate kDOseen/kDFD is given bykDOseen10.
54NkDFDZimm theory predicts that ν = 1/3 in poor solvent where polymers collapse into
compact conformations. Therefore,kDOseen 0.54 N 2 3FDkD

Comparison between theory andsimulation• Note: the units of g and kD are different. The former has the units of s-1 and thelatter has units of mol-1m3s-1. Therefore, direct comparison is difficult.
• However, we believe that our theoretical model can capture an essence of HI
effects on membrane self-assembly.

まとめ• 複雑な膜形成プロセスの、ほんの一部分だ
けかもしれませんが、その背景にある物理化学的なエッセンスを理論的に記述することができたので、個人的に気に入っている仕事です。（既にある理論を組み合わせただけですが、、、）• このような解析は、モデルを“粗視化”しないとできない！

最後に

粗視化モデリング・シミュレーションの個人的な考え• 最近、生命系シミュレーションの分野では、 “粗視化”という言葉をよく聞くようになりました。
• そして、ほとんどの論文で、“粗視化 ＝ 自由度を低くし、長時間シミュレーションを可能にする”、としているようです。
• 確かにそうですが、私の中では、“粗視化 ＝複雑な現象から本質的な部分を抜き出す作業”と考えて、日々研究をしています。

良い言葉が出てこないので、偉い方のお言葉をお借りしました。
Tarmer Schlick著 Molecular Modeling and Simulation: An Interdisciplinary
Guide（Springer）からの文章

I often remind my students of Pablo Picasso’s statement on art:
“Art is the lie that helps tell the truth”. This view applies aptly tobiomolecular modeling.
<途中省略>The key in modeling is to develop and apply models that areappropriate for the questions being examined with them.
Make everything as simple as possible, but not simpler.
Albert Einstein
しかし、これが一番難しい。。。

ありがとうございました。