The rebinding effect and reduced
models for ligand binding processes
Marco Sarich
Freie Universität Berlin
DFG Research Center MATHEON
Mathematics for key technologies
10.09.2013
Bivalent docking examples
Ligand: tamoxifen with PEG bridge
Receptor: a-estrogen receptor
Solvent: water
Marco Sarich - Freie Universität Berlin
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Bivalent docking examples
Ligand: tamoxifen with PEG bridge
Receptor: a-estrogen receptor
Solvent: water
Marco Sarich - Freie Universität Berlin
Ligand: bivalent ammonium-ion
Receptor: bivalent rotaxane
Solvent: water & methanol
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Transition rates or transition probabilities
kon
UNBOUND
BOUND
koff
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Intermediate states lead to problems
UNBOUND
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SINGLY BOUND
DOUBLY BOUND
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Partitioning of state space
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Rebinding effect as recrossing problem
REBINDING EFFECT
=
RECROSSING EVENTS
Weber, Fackeldey, 2013.
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Potential energy landscapes
dXt = −∇V (Xt )dt + σdBt
3
7
6
2.5
5
2
4
1.5
3
1
2
0.5
0
−2
1
−1.5
−1
−0.5
0
0.5
1
1.5
Less pronounced rebinding effect
Marco Sarich - Freie Universität Berlin
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0
−2
0
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Strong rebinding effect for all possible boundaries
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Introduce intermediate macro states
Does this solve the problem?
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potential
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0
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x
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Markov State Models
How well does the Markov chain Pij = P[Xt+τ ∈ Aj |Xt ∈ Ai ] approximate
the dynamics?
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potential
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Markov State Models
Lτ
T := Tτ = e ,
T := Tτ = e Lτ ,
Implied timescales
Implied timescales
λi - largest eigenvalues of
λi - largest eigenvalues of
η1
η1
original 17.5267
original 16.5478
17.5267
full partition
full partition 16.5478
1 2
(Lv )(x) = 1 σ ∆v (x) − ∇V (x) · ∇v (x).
(Lv )(x) = 2 σ 2 ∆v (x) − ∇V (x) · ∇v (x).
2
τ
ηi = − τ
ηi = − log λi
log λ
transfer operatori and its approximation.
transfer operator and its approximation.
η2
η3
η4
η2
η3
η4
3.1701 0.9804 0.4524
3.1701
2.9073 0.9804
0.8941 0.4524
0.4006
2.9073 0.8941 0.4006
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potential
Transfer operator:
Transfer operator:
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Discretization error analysis
D = span{1A1 , ..., 1An }
Q - orthogonal projection onto D
P can be considered as projection of the original transfer operator of the
system
QT : D → D,
T := Tτ = e Lτ ,
Marco Sarich - Freie Universität Berlin
(Lv )(x) =
1 2
σ ∆v (x) − ∇V (x) · ∇v (x).
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Projection error of eigenvectors
T :
λ:
T : self-adjoint transfer operator (reversible dynamics)
T : self-adjoint transfer operator (reversible dynamics)
λ : some eigenvalue
self-adjointλtransfer
(reversible dynamics)
: some operator
eigenvalue
u : a corresponding normalized eigenvector
some eigenvalue
u : a corresponding normalized eigenvector
λ1 : largest non-trivial eigenvalue of T (λ1 < 1)
a corresponding
normalized
eigenvector
λ1 : largest
non-trivial
eigenvalue of T (λ1 < 1)
Q : orthogonal projection onto some subspace D
largest non-trivial
eigenvalue
of T (λ1onto
< 1)some subspace D
Q : orthogonal
projection
δ = Qu − u projection error of eigenvector
orthogonal δprojection
some subspace
= Qu − onto
u projection
error ofDeigenvector
u:
λ1 :
Q:
Then δ = Qu − u projection error of eigenvector
Then
QT has an eigenvalue λ̂ with
Then
QT has an eigenvalue λ̂ with
QT has an eigenvalue λ̂ with
|λ − λ̂| ≤ 2λ1 δ.
|λ − λ̂| ≤ 2λ1 δ.
|λ − λ̂| ≤ 2λ1 δ.
Sarich, Schütte. Comm. Math. Sci., 2012.
Marco Sarich - Freie Universität Berlin
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τ
ηi = −
Projection
log λierror for the example
λi - largest eigenvalues of transfer operator or its projection
original
full partition
η1
η2
η3
η4
17.5267
16.5478
3.1701
2.9073
0.9804
0.8941
0.4524
0.4006
1.5
1
0.5
0
−0.5
−1
−1.5
Marco Sarich - Freie Universität Berlin
0
2
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The core set approach
Use instead of a full partition....
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The core set approach
...so called core sets by cutting out a
transition region.
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The core set approach
C4
C2
C7
x
C5
C1
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C3
C6
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The core set approach and committors
qi (x) = P[process hits Ci next | X0 = x]
C4
C2
C7
x
C5
C1
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C3
C6
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The benefit of core sets
Projection of transfer operator QT onto D = span{q1 , ...qn } leads to
matrix P = T̂ M −1 with
Mij = P[after time t process will hit next Cj |
at time t process came last from Ci ]
and
T̂ij = P[after time t + τ process will hit next Cj |
at time t process came last from Ci ].
A2
C2
t+τ
t+τ
t
A1
Marco Sarich - Freie Universität Berlin
t
C1
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The benefit of core sets
...and it can be proven to be more accurate!
ρ
ρ ,
u
−
Q
u
≤
u
−
Q
u
−
(u
−
Q
u)
+
u − Qcores u ≤cores
u − Qfull u − (u
+ ,C
full − Qfull u) C full
η
η
ρE≤
ρ ≤ max
(C cE
)|]x [τ (C c )|]
x [τmax
η timescale
implied timescale
belonging
to u
η implied
belonging
to u
Cthat
region
that
is cut out
C region
is cut
out
x∈C
x∈C
Sarich, Schütte. Comm. Math. Sci., 2012.
Marco Sarich - Freie Universität Berlin
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The benefit of core sets
removing the error that comes from
approximation by a stepfunction in
the region that is cut out
ρ
ρ ,
u
−
Q
u
≤
u
−
Q
u
−
(u
−
Q
u)
+
u − Qcores u ≤cores
u − Qfull u − (u
+ ,C
full − Qfull u) C full
η
η
ρE≤
ρ ≤ max
(C cE
)|]x [τ (C c )|]
x [τmax
η timescale
implied timescale
belonging
to u
η implied
belonging
to u
Cthat
region
that
is cut out
C region
is cut
out
x∈C
Marco Sarich - Freie Universität Berlin
x∈C
ratio is small if the region that is cut
out is left on a much faster timescale
than the timescale of interest
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The benefit of core sets
Example
original
core sets
full partition
η1
η2
η3
η4
17.5267
17.3298
16.5478
3.1701
3.1332
2.9073
0.9804
0.9690
0.8941
0.4524
0.4430
0.4006
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
0
2
4
6
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−1.5
0
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What if we use no intermediate states?
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A „soft“ partitioning into bound and unbound state
1
0.8
0.6
0.4
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−0.2
−0.4
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The two committor functions define affiliations to the bound
and unbound state.
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Very accurate approximation = no rebinding effect
1.5
1
0.5
0
−0.5
−1
−1.5
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Approximation of the eigenvector by the committors with respect to two macro states.
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Very accurate approximation = no rebinding effect
Example
original
2 core sets
9 core sets
full partition
Marco Sarich - Freie Universität Berlin
T1
T2
T3
T4
17.5267
17.5043
17.3298
16.5478
3.1701
3.1332
2.9073
0.9804
0.9690
0.8941
0.4524
0.4430
0.4006
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Building a finer model
Analyzing the transition behaviour after discretization,
for example using TPT (transition path theory).
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A multilevel/multigrid approach
Using a multigrid
Let Cj = {C1j , ..., Cnj j } be an increasing sequence of core set
discretizations, i.e.
Ci ⊂ Cj for all j ≥ i.
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Iterative Space Construction
Level 1: D1 = span{q1 , ..., qn }, qi usual committors.
Level k + 1: Dk+1 = D̂k+1 ∩ Dk⊥ , where D̂k+1 is the usual committor
space for the cores on level k + 1.
Total space for projection with m levels: D = D1 + ... + Dm
0
2
4
6
8
10
Level 1
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Level 2
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Level 3
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A multilevel/multigrid approach
Using a multigrid
Let Cj = {C1j , ..., Cnj j } be an increasing sequence of core set
discretizations, i.e.
Ci ⊂ Cj for all j ≥ i.
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2
1
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Iterative Space Construction
Level 1: D1 = span{q1 , ..., qn }, qi usual committors.
Level k + 1: Dk+1 = D̂k+1 ∩ Dk⊥ , where D̂k+1 is the usual committor
space
the cores on level k + 1.
Using for
a multigrid
j
+ core
... + set
Dm
Total
with
m levels:sequence
D = D1 of
= {C1for
, ...,projection
Cnj j } be an
increasing
Let Cj space
discretizations, i.e.
Level 1
Level 2
Level 3
Ci ⊂ Cj for all j ≥ i.
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Iterative Space Construction
Level 1: D1 = span{q1 , ..., qn }, qi usual committors.
Level k + 1: Dk+1 = D̂k+1 ∩ Dk⊥ , where D̂k+1 is the usual committor
space for the cores on level k + 1.
Total space for projection with m levels: D = D1 + ... + Dm
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Computation via transition counts
Computing the model
A matrix representation P = T̂ M −1 can again be computed from
stochastic quantities.
Assume Ci was first introduced on level k and Cj was first introduced on
level l, then
Mij = P[hit Cj next on level l | came from Ci on level k].
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No rebinding / recrossing along multiple timescales
1
1
0.9
0.8
0.8
0.6
0.7
0.6
0.4
0.5
0.2
0.4
0.3
0
0.2
−0.2
0.1
0
0
2
4
6
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10
12
−0.4
Usual committors for 4 sets.
0
2
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Mutilevel committors for 4 sets
and 2 levels.
Example
original
9 cores multigrid
9 core sets
full partition
2 core sets
Marco Sarich - Freie Universität Berlin
T1
T2
T3
T4
17.5267
17.5043
17.3298
16.5478
17.5043
3.1701
3.1579
3.1332
2.9073
-
0.9804
0.9703
0.9690
0.8941
-
0.4524
0.4441
0.4430
0.4006
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References
Ch. Schütte and M. Sarich. Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis,
Algorithmic Approaches. Courant Lecture Notes, 24. American Mathematical Society, 2013.
M. Weber, K. Fackeldey. Computing the Minimal Rebinding Effect Included in a Given Kinetics. Submitted to
MMS, 2013.
M. Sarich and Ch. Schütte. Approximating Selected Non-dominant Timescales by Markov State Models. Comm.
Math. Sci., 2012.
N. Djurdjevac, M. Sarich, and Ch. Schütte. Estimating the Eigenvalue Error of Markov State Models. Multiscale
Modeling and Simulation, 10(1): 61–81, 2012.
M. Weber, A. Bujotzek, R. Haag: Quantifying the rebinding effect in multivalent chemical ligand-receptor systems. Journal of Chemical Physics, 137(5):054111, 2012.
C. Schütte, F. Noé, J. Lu, M. Sarich, and E. Vanden-Eijnden. Markov state models based on milestoning. J.
Chem. Phys, 134 (20), 2011.
M. Sarich, F. Noe, and Ch. Schütte. On the approximation quality of markov state models. Multiscale Modeling
and Simulation, 8(4): 1154–1177, 2010.
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