A statistical model for antibody-antigen binding
We discuss a statistical model for antibody-antigen binding. The two macromolecules are assumed to be linked by a number of relatively weak bonds (or groups of correlated bonds) that are assumed to open and close statistically. We use the model for a preliminary analysis of experiments performed in...
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Інститут фізики конденсованих систем НАН України
1999
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| Zitieren: | A statistical model for antibody-antigen binding / S. Katletz, U.M. Titulaer // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 361-368. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1203982017-06-13T03:03:18Z A statistical model for antibody-antigen binding Katletz, S. Titulaer, U.M. We discuss a statistical model for antibody-antigen binding. The two macromolecules are assumed to be linked by a number of relatively weak bonds (or groups of correlated bonds) that are assumed to open and close statistically. We use the model for a preliminary analysis of experiments performed in the Institute of Biophysics at the Johannes Kepler University. In these experiments the two molecules are brought into contact using an atomic force microscope; then a prescribed time dependent force is applied to the bond and the distribution of times needed to pull the molecules completely apart is measured. This quantity is calculated with our model; its dependence on the model parameters (binding free energies, number of groups of correlated elementary bonds, force dependence of the binding free energy) is determined. Обговорюється статистична модель, яка описує зв’язування антитіло-антиген. При цьому вважається, що дві макромолекули можуть поєднуватись через набір відносно слабих зв’язків (чи груп скорельованих зв’язків), що відкриваються і закриваються статистично. Ця модель використовується для попереднього аналізу експериментів, виконаних в Інституті біофізики Університету Йогана Кеплера. У цих експериментах дві молекули приводились у контакт, використовуючи атомної сили мікроскоп, а потім прикладалася певна залежна від часу сила до зв’язку і вимірювався розподіл часів, необхідних для повного розділення молекул. Ця характеристика розраховується з використанням запропонованої моделі; знайдена її залежність від модельних параметрів (вільних енергій зв’язування, числа груп скорельованих елементарних зв’язків, залежності вільної енергії зв’язування від сили). 1999 A statistical model for antibody-antigen binding / S. Katletz, U.M. Titulaer // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 361-368. — Бібліогр.: 5 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.361 PACS: 87.15.B, 87.80, 82.20.M, 05.20, 05.40 http://dspace.nbuv.gov.ua/handle/123456789/120398 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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We discuss a statistical model for antibody-antigen binding. The two macromolecules are assumed to be linked by a number of relatively weak bonds
(or groups of correlated bonds) that are assumed to open and close statistically. We use the model for a preliminary analysis of experiments performed in the Institute of Biophysics at the Johannes Kepler University. In
these experiments the two molecules are brought into contact using an
atomic force microscope; then a prescribed time dependent force is applied to the bond and the distribution of times needed to pull the molecules
completely apart is measured. This quantity is calculated with our model;
its dependence on the model parameters (binding free energies, number
of groups of correlated elementary bonds, force dependence of the binding
free energy) is determined. |
| author |
Katletz, S. Titulaer, U.M. |
| spellingShingle |
Katletz, S. Titulaer, U.M. A statistical model for antibody-antigen binding Condensed Matter Physics |
| author_facet |
Katletz, S. Titulaer, U.M. |
| author_sort |
Katletz, S. |
| title |
A statistical model for antibody-antigen binding |
| title_short |
A statistical model for antibody-antigen binding |
| title_full |
A statistical model for antibody-antigen binding |
| title_fullStr |
A statistical model for antibody-antigen binding |
| title_full_unstemmed |
A statistical model for antibody-antigen binding |
| title_sort |
statistical model for antibody-antigen binding |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120398 |
| citation_txt |
A statistical model for antibody-antigen binding / S. Katletz, U.M. Titulaer // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 361-368. — Бібліогр.: 5 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT katletzs astatisticalmodelforantibodyantigenbinding AT titulaerum astatisticalmodelforantibodyantigenbinding AT katletzs statisticalmodelforantibodyantigenbinding AT titulaerum statisticalmodelforantibodyantigenbinding |
| first_indexed |
2025-07-08T17:48:38Z |
| last_indexed |
2025-07-08T17:48:38Z |
| _version_ |
1837101929108340736 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 361–368
A statistical model for antibody-antigen
binding
S.Katletz, U.M.Titulaer
Institute for Theoretical Physics, Johannes Kepler University
Altenbergerstrasse 69, Linz, Austria
Received July 6, 1998
We discuss a statistical model for antibody-antigen binding. The two macro-
molecules are assumed to be linked by a number of relatively weak bonds
(or groups of correlated bonds) that are assumed to open and close sta-
tistically. We use the model for a preliminary analysis of experiments per-
formed in the Institute of Biophysics at the Johannes Kepler University. In
these experiments the two molecules are brought into contact using an
atomic force microscope; then a prescribed time dependent force is ap-
plied to the bond and the distribution of times needed to pull the molecules
completely apart is measured. This quantity is calculated with our model;
its dependence on the model parameters (binding free energies, number
of groups of correlated elementary bonds, force dependence of the binding
free energy) is determined.
Key words: random walk, one step process, first passage time,
dissociation time, antibody-antigen binding
PACS: 87.15.B, 87.80, 82.20.M, 05.20, 05.40
1. Chemical reaction as a random walk with absorbing barrier
In equilibrium statistical physics, the mass action law gives a relation between
the reaction rates and the concentrations of reactants and products in a chemical
reaction. The equilibrium constant involved is a function of the free energy difference
between initial and final states and the temperature alone. This theory, however,
cannot predict the kinetics of a reaction, since that is determined by the reaction
path and by the potential energy barrier between the equilibrium states. Therefore
one has to use non-equilibrium thermodynamics and introduce a model for the
reaction in order to calculate the dissociation rate.
Montroll and Shuler[1] investigated a model for an uni-molecular reaction of the
type
M + AB ⇀↽ M+ AB
∗,
AB
∗ ⇀ A+ B, (1)
c© S.Katletz, U.M.Titulaer 361
S.Katletz, U.M.Titulaer
0
1
2
N−1
N
Figure 1. Potential energy and energy levels
where M represents the heat bath molecules (the solvent), AB the reactant molecu-
les, AB∗ in their excited states and A+B the dissociation products. They assumed
that the potential energy between the molecules A and B is harmonic, the energy
states are therefore En = hν(n + 1
2
). During the collision between the solvent and
reactant molecules energy is transferred and transitions between the energy states
occur (see figure 1). Now an absorbing barrier is introduced, i.e. when a molecule
AB∗ reaches the N-th energy state it dissociates and is removed (there are no back
reactions).
We denote by xn the probability that a molecule is in the nth state. This quantity
obeys the master equation
dxn
dt
=
∑
m
Wnmxm −Wmnxn
=
∑
m
[
Wnm − δnm
∑
l
Wln
]
xm
=
∑
m
Anmxm (2)
with
Anm = Wnm − δnm
∑
l
Wln . (3)
Wnm is the transition probability per unit time for a transition from level m to
n. In the original model, quantum mechanical perturbation theory for a harmonic
oscillator in a heat bath yields the expression
Wn,n+1 = eβEn+1(n + 1)W0,1 ,
Wn+1,n = eβEn(n + 1)W0,1 . (4)
Note that only nearest neighbour transition can occur; we therefore have a one-step
Markov process.
362
Antibody-antigen binding
We want to describe the dissociation of biomolecules, a reaction where a large
number of small bonds must be broken. We identify the number N with the number
of bonds (not necessarily the total number of bonds in the case that a certain
number of bonds form a group and can only open and close collectively – then
N is the number of groups). EN is the dissociation energy and the subscript n
becomes the number of open bonds (in the ground state 0, in the dissociated state
the total number of bonds or groups). Therefore we have to modify the transition
probabilities: the probability that a bond is reestablished, Wn,n+1 is proportional to
the number of open bonds, Wn+1,n, the probability that a bond breaks, proportional
to the number of closed bonds. Allowing for an external force we are led to the
following transition probabilities:
Wn,n+1 = eβ(En+1+
Fλ
N−n−1
)(n + 1)W0,1 ,
Wn+1,n = eβ(En+
Fλ
N−n
)(N − n)W0,1 . (5)
F is the total force trying to tear the moleculeAB apart and is assumed to be equally
distributed over the bonds or groups of bonds; λ is the proportionality constant in
the dependence of the free energy on the force.
Note that by (5) detailed balance is satisfied when taking the degeneracy of each
state into account:
gmWnmx
e
m = gnWmnx
e
n (6)
with gn =
(
N
n
)
and xe
n the probability distribution in equilibrium
xe
n ∝ e−β(En+
Fλ
N−n
). (7)
The first passage time is the time a molecule needs to reach the absorbing barrier
for the first time. The distribution of the first passage times is proportional to the
number of molecules passing this barrier at a certain time and is given by
p(t) = WN,N−1xN−1(t). (8)
2. Numerical solution
First we transform the matrix of our system of differential equations into a
symmetrical one. The transformation
xi = biyi (9)
with constant coefficients bi yields the recurrence relation
bi =
√
ai,i−1
ai−1,i
bi−1 . (10)
We choose b0 = 1; the elements of B are then given by
Bi,i−1 = ai,i−1
bi−1
bi
=
√
ai−1,iai,i−1, (11)
363
S.Katletz, U.M.Titulaer
which is obviously symmetric. The solution of the new system of differential equa-
tions is given by
~y(t) =
∑
αi
~ξie
λit, (12)
where ~ξi is the eigenvector to the eigenvalue λi satisfying
B · ~ξi = λi
~ξi . (13)
The eigenvectors ~ξi are orthogonal and can be normalized. Then the linear coeffi-
cients αi are determined by the initial values of ~x and hence ~y:
~y(0) =
∑
αi
~ξi , (14)
αi = ~y(0) · ~ξi . (15)
As an initial distribution we choose the Boltzmann distribution,
xn(0) =
exp(−En/kBT )
∑
exp(−En/kBT )
. (16)
Using (8), the distribution of first passage times is given by
p(t) = WN,N−1bN−1yN−1(t)
= WN,N−1bN−1
∑
αiξi[N−1]e
λit, (17)
where ξi[N−1] denotes the N − 1th component of ~ξi.
In case of a time dependent force, (12) is applied for small time interval ∆t
during which the force can be assumed to be constant. The solution ~y(t + ∆t) is
taken as the new initial value with a new force and a new transition matrix B.
2.1. Zero force
The method described above was used to solve the differential equation for var-
ious total energies EN and total number of bond groups N . The mean first passage
time when there is no force
τ0 =
∫
p(t)tdt, (18)
corresponds to the inverse of the dissociation rate 1/koff which was measured exper-
imentally [2]. Thus the factor W01 can be determined.
2.2. Constant force
Figure 3 shows the ratio τ/τ0 of mean first passage times. Its dependence on the
number of bonds and on the applied force is shown. The mean first passage time
decreases exponentially with the applied force, a slope being determined by the total
number of bonds.
364
Antibody-antigen binding
2 4 6 8 10 12 14 16 18 20
N
10
−1
10
0
10
1
10
2
10
3
10
4
τ0 [a.u.]
E/kBT=5
E/kBT=10
E/kBT=15
E/kBT=20
E/kBT=25
E/kBT=30
E/kBT=50
Figure 2. τ0 as a function of N and E/kBT
0 2 4 6 8 10 12 14 16 18 20
N
0
0.2
0.4
0.6
0.8
τ/τ0
E/kBT=20
Fλ/kBT=5
Fλ/kBT=10
Fλ/kBT=15
Fλ/kBT=20
Fλ/kBT=25
Fλ/kBT=30
Fλ/kBT=40
0 10 20 30 40 50
Fλ/kBT
10
−11
10
−9
10
−7
10
−5
10
−3
10
−1
τ/τ0
E/kBT=20
N=2
N=4
N=6
N=8
N=10
N=14
N=20
Figure 3. The mean first passage time at constant forces
365
S.Katletz, U.M.Titulaer
0 2 4 6 8 10 12 14 16 18 20
N
0
0.2
0.4
0.6
0.8
τ/τ0
E/kBT=20
ω=10
ω=40
ω=70
ω=100
ω=130
ω=160
ω=190
0 200 400 600 800 1000
ω
10
−2
10
−1
10
0
τ/τ0
E/kBT=20
N=4
N=8
N=12
N=16
N=20
Figure 4. The mean first passage time at linearly increasing force
10 100 1000
ω
0
50
100
150
fu
E/kBT=25
N=4
N=6
N=8
N=12
Figure 5. Mean rupture force
0 100 200
F [pN]
0
0.01
0.02
0.03
p
ω=300
N=4
3 Hz
N=6
N=8
Figure 6. Distribution of rupture
forces
2.3. Linearly increasing force
Finally a force varying according to
F (t)λ
kBT
= ω
t
τ0
(19)
was inserted into the differential equation. Figure 4 shows the dependence of the
mean first passage time on the total number of bonds and the pulling velocity.
We are also able to identify the distribution of rupture forces with the distribution
of first passage times if we draw F (t) instead of t. Furthermore we can compare
these results with the experiments performed at the Institute of Biophysics at the
University of Linz [3,4].
Figure 5 shows that the mean rupture force increases roughly logarithmically
with the pulling velocity, a fact also observed in the experiments. The slope is
366
Antibody-antigen binding
determined by the number of groups.
Finally we tried to fit the experimental rupture force distributions (antibody
HyHel 8, cantilever with spring constant 30pN/nm, scanning frequency 3Hz) with
our model as shown in figure 6. With our choice of λ = 1.2nm and ω = 300 the
system with 4 groups seems to fit the data best.
3. Conclusions
Though there is still some disagreement between experimental and theoretical
distributions of the rupture force, the dependence of the mean rupture force on the
pulling velocity shows that the model is capable of describing the experiment on the
right time scale. The difference may be due to the nonlinear increase of the force
that actually acted on the binding [5]. In a further investigation this nonlinear force
increase will be used, allowing a better way of determining the force constant λ.
Another modification that will be necessary is to relax the condition that a
group of bonds open or close together instantaneously. Although a high degree of
correlation is plausible on biophysical grounds, a complete correlation is certainly
too rough an approximation to obtain quantitative agreement with the experiments.
Acknowledgements: It is a pleasure to thank Prof. Hansgeorg Schindler, Dr. Pe-
ter Hinterdorfer and Ms Anneliese Raab (Institute of Biophysics) for very fruitful
discussions.
References
1. Montroll E.W., Shuler K.E. The application of the theory of stochastic processes to
chemical kinetics. Advances in Chemical Physics 1 (I. Prigogine ed., Interscience, New
York 1958), p. 361–399.
2. Mohan S., Smith-Gill S.J. Conformational flexibility and cognitive properties of cross-
reactive high affinity antibodies. // to be published.
3. Hinterdorfer P., Baumgartner W., Gruber H.J., Schilcher K., Schindler H. Detection
and localization of individual antibody-antigen recognition events by atomic force mi-
croscopy. // Proc. Natl. Acad. Sci. USA, 1996, vol. 93, p. 3477–3481.
4. Hinterdorfer P., Raab A., Badt D., Smith-Gill S.J., Schindler H. Force spectroscopy of
antibody-antigen recognition measured by scanning force microscopy. // Biophys. J.,
1998, vol. 74 (2), p. A 186.
5. The force vs. extension dependence of the spacer molecule PEG, which connects the
antibody to the cantilever of the AFM, agrees very well with the freely joint chain
model. In this model the extension is given by the Langevin equation plus a linear
dependence due to the cantilever.
x = Nl
(
coth
FL
kBT
− kBT
FL
)
+
F
k
(20)
L is the Kuhn length, Nl the contour length of the polymer, k the spring constant of
the cantilever. See for example: Eyring H., Henderson D., Jones Stover B., Eyring E.M.
Statistical Mechanics and Dynamics. New York, John Wiley & Sons, 1964.
367
S.Katletz, U.M.Titulaer
Статистична модель зв’язування антитіло-антиген
С.Катлєц, У.Тітуляр
Інститут теоретичної фізики, Університет Йогана Кеплера
Австрія, Лінц, Алтенбергштрассе, 69
Отримано 6 липня 1998 р.
Обговорюється статистична модель, яка описує зв’язування анти-
тіло-антиген. При цьому вважається, що дві макромолекули можуть
поєднуватись через набір відносно слабих зв’язків (чи груп скоре-
льованих зв’язків), що відкриваються і закриваються статистично. Ця
модель використовується для попереднього аналізу експериментів,
виконаних в Інституті біофізики Університету Йогана Кеплера. У цих
експериментах дві молекули приводились у контакт, використовую-
чи атомної сили мікроскоп, а потім прикладалася певна залежна від
часу сила до зв’язку і вимірювався розподіл часів, необхідних для по-
вного розділення молекул. Ця характеристика розраховується з ви-
користанням запропонованої моделі; знайдена її залежність від мо-
дельних параметрів (вільних енергій зв’язування, числа груп скоре-
льованих елементарних зв’язків, залежності вільної енергії зв’язу-
вання від сили).
Ключові слова: вільні блукання, однокрокові процеси, час першої
події, час дисоціації, антитіло-антиген зв’язування
PACS: 87.15.B, 87.80, 82.20.M, 05.20, 05.40
368
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