An alternative to the "big molecules" view of proteins is the "small things" view in which protein have a shape and material properties. This talk is about investigating these properties.
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
The stuff that proteins are made of
1. The stuff that proteins are made of:
physical properties of folded peptide chains
Konrad Hinsen
Centre de Biophysique Moléculaire, Orléans (France)
Synchrotron SOLEIL, Saint Aubin (France)
4. Composite materials
By Lionel Allorge - Own work, CC BY-SA 3.0
https://commons.wikimedia.org/w/index.php?curid=25507982
5. The stuff that proteins are made of
“Folded peptide chains” as a construction material
Start from macroscopic properties (density,
elasticity, …)
Add meso-/microscopic details:
heterogeneity (pockets, …)
local anisotropy
chain structure
6. Motivation
For modelling very large
protein assemblies:
virus capsids, ribosomes, …
Coarse-graining has its limits.
At larger scales, continuum
models are more reasonable.
By Thomas Splettstoesser (www.scistyle.com)
(Own work) CC BY-SA 4.0
https://commons.wikimedia.org/wiki/File:Zika_virus_capsid.png
8. Elastic network model (ENM)
Coarse-grained molecule: only Cα atoms
Springs between all atom pairs
K. Hinsen, Proteins 33, 417-429 (1998)
4.2. E€ective potential well
In a study of domain motions in large proteins
by normal mode analysis [27,28], it was found that
domain motions can be reproduced using a simple
harmonic potential of the form
U…R1; . . . ; RN † ˆ
X
all pairs a;b
Uab…Ra À Rb† …10†
with the pair potential
Uab…r† ˆ k R
…0†
ab
rj j
À R
…0†
ab
9.
10.
11.
12.
13.
14. 2
: …11†
Here and in the following xa ˆ Ra À Req
a are the
particle displacements with respect to the equilib-
rium position, and R
…0†
ab ˆ Req
a À Req
b is the pair
distance vector in the stable equilibrium con®gu-
mical Physics 261 (2000) 25±37
tive potential well
udy of domain motions in large proteins
l mode analysis [27,28], it was found that
motions can be reproduced using a simple
potential of the form
; RN † ˆ
X
all pairs a;b
Uab…Ra À Rb† …10†
pair potential
k R
…0†
ab
rj j
À R
…0†
ab
15.
16.
17.
18.
19.
20. 2
: …11†
in the following xa ˆ Ra À Req
a are the
isplacements with respect to the equilib-
…0† eq eq
000) 25±37
K. Hinsen, A.J. Petrescu, S. Dellerue,
M.C. Bellissent-Funel G.R. Kneller,
Chem. Phys. 261, 25-37 (2000)
21. From ENMs to elastic media
Elasticity at the macroscopic scale:
continuous media
Material parameters: density , elastic tensor
Question: what’s the smallest length scale
on which proteins can be described in this way?
⇢ ij,kl
23. Experimental data:
Elastic constants of lysozyme crystals
Observation of all the components of elastic constants using tetragonal hen egg-white lysozyme
crystals dehydrated at 42% relative humidity
H. Koizumi, M. Tachibana, and K. Kojima
Graduate School of Integrated Science, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan
͑Received 11 August 2004; revised manuscript received 23 January 2006; published 10 April 2006͒
Success in measuring transverse sound velocity allowed us to determine, for the first time, all six elastic
constants of a protein crystal. An ultrasonic pulse-echo method was used to perform sound velocity measure-
ments on tetragonal hen egg-white ͑HEW͒ lysozyme crystals that were partially dehydrated at 42% relative
humidity. The measurements were performed using the ͑110͒, ͑101͒, and ͑001͒ crystallographic faces. Thus, all
six elastic constants of the dehydrated tetragonal HEW lysozyme crystals were determined: C11=C22
=12.44 GPa, C12=7.03 GPa, C13=C23=8.36 GPa, C33=12.79 GPa, C44=C55=2.97 GPa, and C66=2.63 GPa.
In addition, for the hydrated crystals, the longitudinal sound velocities along the ͓110͔ direction and the
direction normal to the ͑101͒ face were measured. From these results, all the components of elastic constants
in the hydrated crystals were extrapolated.
DOI: 10.1103/PhysRevE.73.041910 PACS number͑s͒: 87.15.Ϫv, 62.20.Ϫx, 62.30.ϩd, 62.65.ϩk
I. INTRODUCTION
In order to advance research on protein molecules and
protein crystals, it is necessary to grow highly perfect crys-
tals. This is because the defect structures of the crystals
hinder the determination of three-dimensional protein struc-
tures using x-ray diffraction and neutron diffraction methods.
To determine all the components of elastic constants in
protein crystals, both the longitudinal and transverse sound
velocities need to be measured. In the pulse-echo method, in
order to generate a transverse ultrasonic wave in the crystal,
a transducer must be in close contact with the crystal. How-
ever, since hydrated protein crystals are fragile, it was diffi-
PHYSICAL REVIEW E 73, 041910 ͑2006͒
PHYSICAL REVIEW E 89, 012714 (2014)
Elastic constants in orthorhombic hen egg-white lysozyme crystals
N. Kitajima,1
S. Tsukashima,2
D. Fujii,2
M. Tachibana,2
H. Koizumi,3
K. Wako,4,*
and K. Kojima4
1
Citizen Holdings Company, Ltd, 840, Shimotomi, Tokorozawa, Saitama 359-8511, Japan
2
Graduate School of Nanobioscience, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan
3
Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
4
Department of Education, Yokohama Soei University, 1 Miho-cho, Midori-ku Yokohama 226-0015, Japan
(Received 10 March 2013; published 21 January 2014)
The ultrasonic sound velocities of cross-linked orthorhombic hen egg-white lysozyme (HEWL) crystals,
including a large amount of water in the crystal, were measured using an ultrasonic pulse-echo method. As a
result, seven elastic constants of orthorhombic crystals were observed to be C11 = 5.24 GPa, C22 = 4.87 GPa,
C12 = 4.02 GPa, C33 = 5.23 GPa, C44 = 0.30 GPa, C55 = 0.40 GPa, and C66 = 0.43 GPa, respectively. However,
24. ENMs for protein crystals
Apply ENM to an infinite perfect crystal and
compute normal modes.
Compute the modes for a continuum model and
compare.
Macroscopic validation: calculate elastic
constants and compare to experiment.
Microscopic validation: calculate atomic
fluctuations and compare to experiment.
K. Hinsen, Bioinformatics 24, 521 (2008)
29. Slow diffusion
Brownian dynamics
Effective well potential obtained by scaling the
local minimum potential
Independent of fast vibrations
➡ Ornstein-Uhlenbeck process, Brownian modes
@
@t
P = kBTr · 1
· rP + r · 1
· K · (r R)P
G.R. Kneller, Chem. Phys. 261, 1-24 (2000)
K. Hinsen, A.J. Petrescu, S. Dellerue, M.C. Bellissent-Funel G.R. Kneller,
Chem. Phys. 261, 25-37 (2000)
30. The normal mode family
the central approximation of the method, which therefore deserves a mor
n.
monic potential well has the form1
U(r) =
1
2
(r R) · K(R) · (r R) ,
is a 3N-dimensional vector (N is the number of atoms) describing
tion at the center of the well and r is an equally 3N-dimensional vector re
nt conformation. The symmetric and positive semidefinite matrix K des
ourselves to harmonic potentials in Cartesian coordinates. Other coordinate can be used as wel
or numerical applications. Note that a potential that is harmonic in one coordinate set is in general
dinates.
Energetic modes: eigenvalues of
➡ force constants
Vibrational modes: eigenvalues of
➡ vibrational frequencies
Brownian modes: eigenvalues of
➡ relaxation rates
Harmonic potential:
K
1/2
· K · 1/2
M 1/2
· K · M 1/2
31. Friction constants
0 200 400 600 800
Surrounding protein density [amu/nm
3
]
0
10
20
30
Frictionconstant[1000amu/ps]
measured friction constants
linear fit
extracted from MD simulations
32. Intermediate scattering function
0 200 400 600 800 1000
Time [ps]
0
0.2
0.4
0.6
0.8
1
Finc
(q,t)
Molecular Dynamics
Brownian modes + vibrational term
q = 10 nm
-1
q = 15 nm
-1
q = 25 nm
-1
q = 20 nm
-1
which read explicitly
Fcoh…q; t† ˆ
ˆ
a;b
ba;cohbb;coh exp iqT
Á Rb…t†
À Á
exp
À
À iqT
Á Ra…0†
Á
;
Finc…q; t† ˆ
ˆ
a
b2
a;inc exp iqT
Á Ra…t†
À Á
exp
À
À iqT
Á Ra…0†
Á
:
Here and in the following Greek indices label atoms, ba;coh is the coherent scattering length o
its incoherent scattering length, and Ra…t† its position operator in the Heisenberg representat
the scattering lengths can be found in standard books on neutron scattering [1,2]. The brack
and (4) denote quantum statistical averages, and the superscript T of a vector indicates a tr
should be noted that Fcoh…q; t† probes collective motions, whereas Finc…q; t† probes only
motions. The quantum correlation functions can be replaced by their classical counterparts i
system can be described by classical mechanics and if recoil e€ects can be neglected [19]. The
classical mechanics is appropriate for an harmonic system if the spacing of the energy
compared to kBT ,
hxn ( kBT:
Here, kB denotes the Boltzmann constant and T, the temperature in Kelvin. Recoil e€ects dep
on the mass of the scattering atom and the potential energy function of the system. For harm
scatterers one obtains a global correction factor exp…hx=2kBT† for the dynamic structu
Therefore the recoil correction can be neglected for harmonic systems if one considers ener
the order of the characteristic frequencies ful®lling (5).
From Eqs. (3) and (4) one obtains two static correlation functions which are frequently
neutron scattering experiments: the static structure factor, S…q† ˆ Fcoh…q; 0†, and the ela
33. Anomalous diffusion
Fast initial but very slow non-exponential long-
time relaxation
Friction constant ➡ memory kernel
G.R. Kneller, K. Hinsen, P. Calligari
J. Chem. Phys. 19, 191101 (2012)
res. 9, b=0.167
a=2.79¥10-4
, t=0.274 ps
100 200 300 400 500
t @psD
0.2
0.4
0.6
0.8
1.0
y@têt;a,bD
res. 29, b=0.12
a=9.905¥10-5
, t=0.117 ps
100 200 300 400 500
t @psD
0.2
0.4
0.6
0.8
1.0
y@têt;a,bD
res. 47, b=0.107
a=2.607¥10-2
, t=17.6 ps
100 200 300 400 500
t @psD
0.2
0.4
0.6
0.8
1.0
y@têt;a,bD
res. 104, b=0.0346
a=5.272¥10-3
, t=7.84 ps
100 200 300 400 500
t @psD
0.2
0.4
0.6
0.8
1.0
y@têt;a,bD
ensemble averages which should exist. For non-
diffusive dynamics, where L is a many-particle
ki operator,21,22
1
2
d⟨[u(t) − u(0)]2
⟩
dt t=0
= −c(1)
(0+), (1)
ticular the short-time diffusion coefficient.
aper, we develop a realistic minimal model for
dynamics of proteins which leads to regular
e Cα-atoms describing both the diffusive short-
s and the relaxation for long times. We assume
escribed by a stationary stochastic process and
correlation function in the form
c(t) = ⟨u2
⟩ψ(t/τ), (2)
the normalized PACF for a dimensionless time
th ψ(0) = 1, and τ 0 sets the time scale. For
we set τ = 1 in the following. To express the
haracter of protein dynamics we write the PACFs
ition of exponential functions,
ψ(t) =
∞
dλ p(λ) exp(−λt), (3)
J. Chem. Phys. 136, 191101 (2012)
λ) is a yet undetermined function fulfilling
λ) = C. The constant C must be chosen such
(λ; β) = 1. We note that limβ → 1sin (πβ) (1 − β)
tion (10) is a necessary and sufficient condition for a
caying PACF with the asymptotic form (5). To con-
) such that the existence of all moments λk and thus
icity of ψ(t) in t = 0 is guaranteed we set
f (λ) = C exp(−βλ). (11)
erly normalized relaxation rate spectrum then reads
p(λ; β) =
λβ−1
ββ
exp(−βλ)
(β)
, (12)
s given by
ψ(t; α, β) =
exp(−αt)
(1 + t/β)β
. (13)
sponding cumulants, which are defined through
c(k)
α,β = (−1)k dk
dtk
ln(ψ(t; α, β))
t=0+
(14)
particularly simple form
Position autocorrelation:
34. To-do list
Normal modes for anomalous diffusion
Link to macroscopic viscoelastic properties:
“visco” ➡ match diffusive modes
“elastic” ➡ match elastic constants
35. A step back towards
smaller length scales:
Coarse-graining peptide chains
43. Macro- and mesoscopic models
Continuous medium approximations become useful
for length scales of ≈ 30 nm.
Intermediate descriptions between the atomic and
the continuum scale are possible.
Friction effects are important for dynamical models
with associated time scales.
A physicist’s dream: one can go a long way with
nothing but harmonic models.
