Topic 1: "Inferential protein structure determination using chemical shifts derived from quantum mechanics" Topic 2: "Hybrid RHF/MP2 geometry optimizations in the Effective Fragment Molecular Orbital Method"

1. Inferential protein structure determination using
chemical shifts derived from quantum mechanics
Anders S. Christensen, Troels E. Linnet, Mikael Borg, Wouter Boomsma, Kresten
Lindorff-Larsen, Thomas Hamelryck, Jan H. Jensen
Hybrid RHF/MP2 geometry optimizations in the
Effective Fragment Molecular Orbital Method
Anders S. Christensen, Casper Steinmann, Dmitri G. Fedorov, Jan H. Jensen
andersx@nano.ku.dk
EFMO-RHF:MP2 geometry
optimizations
Energy of a molecular system in the effective fragment
potential:
EEFMO
=
I
E0
I +
RIJ≤Rresdim
I>J


∆E0
IJ − EPOL
IJ



+
RIJ>Rresdim
I>J
EES
+ EPOL
tot . (1)
In the Frozen Domain approximation:
EEFMO
= E0
F + E0
A + E0
F/A + EPOL
tot , (2)
Figure 1: The geometry of the F layer is frozen during
optimization. The active layer A is optimized the RHF
level, except for the S fragment at MP2.
This gives the EFMO-RHF:MP2 energy:
EEFMO−RHF:MP2
= E0,RHF
F + E0,RHF
A + E0,RHF
F/A (3)
+EPOL
tot + E0,MP2
S∈A , (4)
with the gradient(implemented in GAMESS):
∂EEFMO-RHF:MP2
∂xA
=
∂EEFMO
∂xA
+
∂EMP2
S∈A
∂xA
(5)
Results and discussions
OH
O
O O-
O
O-
O
O-
O
O
O-
HO
OH
O
O
O-
-
O
O
1
4
2
3 3
2
1
4
1
4
3
2
1 2 3
Figure 2: Claisen rearrangement of prephanate to choris-
mate.
Table 1: Electronic energy barrier for the conversion
of prephanate to chorismate in Chorismate Mutase and
the corresponding reaction coordinate for the transition
state using EFMO-RHF:MP2 or ONIOM-RHF:MP2 (from
Steinmann et al).
Method MP2 basis R(TS) Barrier
[kcal/mol]
EFMO 6-31G(d) -0.17 Å 20.95
EFMO cc-pVDZ -0.43 Å 19.21
EFMO cc-pVTZ -0.43 Å 18.34
ONIOM 6-31G(d) 0.13 Å 22.24
ONIOM cc-pVDZ -0.36 Å 19.75
ONIOM cc-pVTZ 0.13 Å 21.79
ONIOM cc-pVQZ 0.13 Å 21.68
Acknowlegements
Collaboration:
Jan H. Jensen (KU/Chem), Casper Steinmann (SDU),
Dimitri G. Fedorov (IAST/Japan)
Inferential protein structure
determination
Calculating protein chemical shifts
Protein chemical shifts can be approximated as:
δ = ∆δBB(φ, ψ, {χn}) + ∆δHB + ∆δRC (6)
where δBB(φ, ψ, {χn}) is a term that depends on the tor-
sion angles, ∆δHB is a sum that depends on the local
hydrogen bonding network, and ∆δRC describes perturba-
tion from ring-currents in nearby aromatic residues. The
individual terms are parametrized from nearly 1,000,000
QM calculations and implemented in the open-source
ProCS program (part of the PHAISTOS frame work).
Sampling protein structures
Protein structures are sampled from a Bayesian posterior
distribution:
p (X| {δexp
i } , I) =
p ({δexp
i } |X, I) p (X|I)
p ({δexp
i } , I)
(7)
which relates the sampled structure, X to a set of mea-
sured experimental chemical shifts, {δexp
i }, and other prior
information, I (e.g. amino acid sequence, etc).
Calculating likelihood
The agreement between a structure X and a set of mea-
sured experimental chemical shifts, {δexp
i }, is calculated
using a Gaussian error model:
p ({δexp
i } |X, {σi}) =
n
i=1











1
2πσ2
i
exp −
(∆δi)2
2σ2
i











, (8)
where ∆δi is the difference between experimental chemical
shifts for the i’th nucleus and the chemical shifts calcu-
lated using Eq. 6. σi denotes the uncertainty of the pre-
dicted value of δi. Our program supports the QM based
ProCS model and the empirical CamShift model.
p (X|I) can be evaluated using a molecular mechanics
force field energy, EFF:
p (X|I) = exp −
EFF
kBT
(9)
Our program supports the OPLS/AA-L force field with a
GB/SA solvent model, and the corse-grained PROFASI
force field.
Results and discussions
Folding of Protein G (56 residues) takes less than a day
with the developed code.
Figure 3: Protein G, from unfolded to folded state. The
blue structure is the experimental NMR structure (PDB:
2OED), and green are snapshots from sampling using the
OPLS-AA/L force field with a GB/SA solvent model and
chemical shifts from CamShift.
Average hydrogen bond length in three proteins
Figure 4: Structures are sampled using the OPLS-AA/L
force field with a GB/SA solvent model and amide proton
chemical shifts from either ProCS or CamShift. Structures
sampled with ProCS show better agreement with exper-
imental data than structures sampled using CamShift or
no chemical shift restraints.
Acknowlegements
Funding:
Novo Nordisk STAR Program
Collaboration and students:
Jan H. Jensen (KU/Chem), Thomas Hamelryck
(KU/Binf), Jens Breinholt (Novo), Kresten Lindorff-
Larsen (KU/NMR), Stephan P. A. Sauer (KU/Chem),
Anders Larsen (KU/Nano), Maher Channir (KU/Nano),
Lars Brathol (KU/Nano), Wouter Boomsma (KU/Binf),
Simon Olsson (KU/Binf), Mikael Borg (KU/Binf)
References
[1] Steinmann C, Fedorov D G, Jensen J H (2010) Effective
fragment molecular orbital method: A merger of the effective
fragment potential and fragment molecular orbital methods. J
Phys Chem A 114:8705-8712.
[2] Hybrid RHF/MP2 geometry optimizations with the Effective
Fragment Molecular Orbital Method (2013) Christensen A S,
Steinmann C, Fedorov D G, Jensen JH arXiv preprint
arXiv:1305.0676
[3] Steinmann C, Fedorov D G, Jensen J H (2013) Mapping
Enzymatic Catalysis using the Effective Fragment Molecular
Orbital Method: Towards all ab initio Biochemistry. PLoS ONE
8(4): e60602.
[4] Christensen A S, Linnet T E, Borg M, Boomsma W,
Lindorf-Larsen K, Hamelryk T, Jensen J H (2013) Protein
structure validation and refinement using amide proton
chemical shifts derived from quantum mechanics PLOS ONE,
accepted arXiv:1305.2164
[5] Boomsma W, et al. (2013) PHAISTOS: A Framework for
Markov Chain Monte Carlo Simulation and Inference of Protein
Structure J Comp Chem (in press, DOI: 10.1002/jcc.23292).
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ESQC, September 2013, Italy