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Modeling
Contents
1.
Introduction
2.
Model
1: ‘Ready’ –
‘Steady‘- cut
2.1
Reaction
kinetics
2.2.
ODEs
2.3
Stimulating temperature dependence for a set of parameters
2.4.
Discussion
3.
Model
2: bind - ‘Ready’- ‘Steady’-
cut
3.1
Reaction
kinetics
3.2
ODE’s
3.3
Different concentrations of Fok_a and Fok_i
3.3.1
Discussion
1.
Introduction
Endonucleases
are restriction enzymes cutting single (ss) - or double (ds) - stranded
DNA at
specific nucleotide sequences. They are found in bacteria and classify
in three
different types of those enzymes, which are categorized in different
groups.
The restriction enzyme of our interest, FokI, belongs to class two,
which means
that it cuts the DNA strand directly after the restriction site.
With
this model we tried to
simulate a
universal enzyme.
Whereas
for
cutting determined substrate is ds DNA, two DNA pieces are the
products, the
measurable output resulting from the reaction.
To
analyze
the process, first the dimerization of the two protein domains needs to
proceed. Therefore models for association (Fig1.), cleavage (Fig.2) and
dissociation (Fig.3) of the different Fok domains, Fok_i and Fok_a, are
introduced. With the help of ODEs (Ordinary differential equations)
simulations
were done, even though a limited set of data is given.
Figure
1:
Association of linker FluA and Dig with DNA and Fok_a and
Fok_i monomers.
2.
After the
accomplishment of the heterodimer,
the
cleavage domain is prepared to cut the DNA.
Figure
2: Cleavage of DNA
double strand
3. Because the unidirectional
cutting process took place, the two DNA
fragments and the restriction enzyme dissociate.
Figure
3: Dissociation of construct after cleavage
2.
Model 1,
‘Ready’-‘Steady’-cut
One
part to
activate the enzyme is the dimerization of the two cutting domains
Fok_i and
Fok_a. Fok_i is linked to FluA (the anticalin binding fluorescein)
compared to
Fok_a, which is linked DigA (the anticalin binding digoxigenin). Here,
we
assume Fok_a and Fok_i dimerize only when the adapter bound to the
tagged DNA
and both DigA and FluA are connected with the linker specific tags
hybrid by
the DNA strand.
As
the two
adapter domains, DigA and FluA, are bound and Fok_i and Fok_a are close
enough
to each other, they can dimerize. In order to get an active enzyme the
Fok_a
/Fok_i heterodimer has to shift into an active state, which is in our
model
called ‘Steady’. But before the enzyme reaches the
active state ‘Steady’, it
takes time to change its conformation. For this reason the enzymes
reach a
condition, which is not yet activated. This step is called
‘Ready’. An enzyme passed
through the step ‘Ready’ and arrived in
‘Steady’ is able to cut a double
stranded DNA.
A
schematized pathway (Fig 4.) shows the sequence of enzyme activation.
Fig.4: Flowchart
model 1
2.1
Reaction
kinetics
The
parts
Fok_i and Fok_a bind DNA at the same time and with the reaction rate k1on and change to the state
‘Ready’,
whereas they dissociate with the rate k1off.
The
enzyme
becomes activated with rate k2_on
and dissociates with the reaction rate k2_off:
After
activation, the enzyme cuts with the rate k3:
2.2
ODEs
All
ODE’s
are derived from the reaction kinetics. The association and
dissociation
constants are analog to the constants in the flow chart above (Fig. 4).
2.3
Simulating temperature dependence for a set of
parameters
Temperature
plays an important role concerning biochemical reactions. It has a high
influence
on the reaction rate. The collision frequency is increased and
molecules have
more thermal energy at higher temperatures.
We
simulate
increased temperature by increased rate constants of enzymes and we
provide
increased dissociation rates. For this reason it is interesting to test
how the
model behaves under different temperatures. Three different cases are
introduced: temperature around optimum, higher temperature than optimum
and
lower temperature than optimum.
The
ODE’s
above are solved numerically; the chosen set of parameters test the
model
corresponding to the above-mentioned circumstances. Later on the
results will
be discussed.
Fig.5:
Temperature around optimum, Model 1
Table1:
Fig.6:
Higher temperature than optimum, Model 1
Table2:
Fig.7:
Lower temperature than optimum, Model 1
Table3:
2.4
Discussion
We
generated ODEs modeling both complex formation and DNA cleavage.
This
process
clearly depends on the temperature of the reaction environment. A
process
mimicking the higher temperature (which leads i.e. to an increased k3 )
is
characterized by faster increasing of the DNA fragments curve and
because of
this, intermediate
products exist
for a shorter time period. In
contrast a process
mimicking lower
temperature, intermediate products are more stable.
At
this
point one has to remark that model 1 is in a manner too much
simplified. To render
the model more realistic, this simulation should consider a limited
temperature
range. Even experiments are done in vitro,
every enzyme cannot work at temperatures too high or too low.
3.
Model 2, bind - ‘Ready’-
‘Steady’- cut
Similar
to
model 1, we developed an extended second model sharing the same basic
concept. In
the second model we additionally considered the formation of Fok
monomer/ DNA
intermediate states.
The
significant difference between both models is the assumption that the
whole
process works slightly more complicated than considered in model 1.
By
enzyme
reactions chemical equations show which educts have to react with each
other to
generate a certain number of products. In our case Fok_a, Fok_i and DNA
represent
educts, whereas the active heterodimer is the final product coming out
of the
reaction. On the way from educts to product interstage products appear
such as DNAFok_a,
DNAFok_i Ready and Steady. However, the likelihood of three parts
colliding
with each other at the same time is quite low. In a first step, two
parts associate
and form an interstage product. In a second or third step the final
product is
formed when distinct interstage products collide with each other. At
this point
one has to remark that the anticalins of the Fok- Monomers are attached
to the
DNA, so that Fok_a has to bind next to Fok_i and vice versa. Other
binding combinations
do not allow dimerization. Hence the central point of this model is the
formation and interaction of all generated interstate products.
As
a
consequence of these considerations, we now consolidated the
assumptions for
our second model.
The
flowchart below shows a schematic presentation of model 2 (Fig. 8).
Fig.8:
Flowchart of model 2
3.1
Reaction kinetics
The
process
of binding Fok_a, Fok_i and DNA is of dynamic nature. Fok_a binds to
the DNA
with the reaction rate k1a_on or a
DNAFok_a complex dissociates with the reaction rate k1a_off.
Likewise
Fok_i binds to the DNA with the reaction rate k1i_on
and the DNAFok_i complex dissociates with the reaction rate k1i_off:
The
complex
DNAFok_a binds Fok_i with reaction rate k2a_on
forming a trimetric complex which dissociates with reaction
rate k2a_off. In a similar fashion,
DNAFok_i
binds Fok_a with the rate k2i_on
and
dissociates with the rate k2i_off.
We
call these states ‘Ready’:
We
now assume
that the Fok_a and Fok_i complexes are initially in the inactive
‘Ready’ state,
which is subsequently converted to the ‘Steady ‘
state. It is activated with
rate k3_on and becomes inactive
with
rate k3_off:
After
activation, the enzyme cuts the DNA substrate with the rate k4. In contrast to the proceeding
steps, this is a unidirectional reaction:
3.2
ODE’s
The
ODE’s
can again be derived from the reaction kinetics and are as follows:
3.3
Different Concentrations of Fok_a and Fok_i
According
to the collision theory, the concentrations of the different reactants
play an
important role for their interactions and also in our special case of
the
formation of an active complex. To
react
with each other, single parts have to collide and the likelihood of
collision
increases with the concentration of every single reactant.
On
the
other hand if the concentrations differ, a disequilibrium arises and the balance of the
reaction is shifted
to one side of the reaction.
The
use of
different amounts of reactants may represent the most important impact
on our
model. As the Parts Fok_a and Fok_i are the basis to start the cleavage
process,
their amounts strongly determine the number of active enzymes.
In
the
following simulation we tested how the process would react if Fok_a
would be in
excess of Fok_i. The increased concentration is simulated by elevated
association
rates. Again, the ODEs are solved for a set of parameters and the
concentrations of the different reaction partners are revealed amongst
the time
course.
Fig.
9: Fok_a and Fok_i in equilibrium, Model 2
Table4:
Chosen set of parameters
Fig.
10: More Fok_a than Fok_i, Model 2
Table5:
Chosen set of parameters
The
following diagram nicely represents the relation between the
concentration of
Fok_a and Fok_i and how the amount of active protein complexes changes
along the
time course. As expected, at a ratio of Fok_a/Fok_i = 1 the highest
concentration of active protein can be observed. Whenever the
concentration equilibrium
shifts to either Fok_a or Fok_i, the amount of active protein
decreases. In vivo, this would
occur if one part is
higher expressed compared to the other part.
.
Fig.
11: Fok_a/Fok_i
Ratio, Model 2
Table6:
Chosen set of parameters
5.
Results
In
addition, it would be interesting to enhance our model
2 with the assumption that Fok_a and Fok_i could also dimerize before
one of
the parts is bound to the DNA via its anticalin.
Based
on the assumption that dimerization of Fok_i/
Fok_a is possible without the need of preceding DNA-binding of DigA or
FluA the
possibility arises that one of the tags necessary to create an active
complex
on the DNA is already occupied by another monomer. This would prevent
cleavage
activity of the preassembled heterodimer because it is not able to bind
both
tags of the target DNA sequence. Eventually, this would lead to an
overall decreased
rate of DNA cleavage.
Based
our considerations, the following equation has
to be added to the reaction kinetics of our model 2 and represents the
dimerization process of Fok_a and Fok_i.
One
has to keep in mind that all the models above are
simplified and do not include all variables influencing the
interactions. Models
always have to be tested if they are close enough to reality and
therefore it
is a necessity to feed a model with realistic data.
Simulations
able to represent in vitro
conditions would have to include tremendous numbers of
physicochemical factors such as solvent properties, salt concentrations
and
structural diversities of the involved molecules. This would require
setup and
computation of enormous equation.
However,
predictions made on the basis of models like
ours can often be used to optimize conditions for simple experimental
setups. For
example, the effects of the lower expression rates of our Fok_a
constructs can
be visualized before the actual experiment is performed.
5.
Literature:
- 1. Gerald
Beste, Frank Schmidt, Thomas Stibora
and Arne Skerra: “Small antibody-like proteins with
prescribed ligand specificities derived from the lipocalin
fold“, Proc. Natl. Acad. Sci. USA, Vol 96, pp. 1898
– 1903, March 1999, Biochemistry
- 2. David Wah, Joel Hirsch, Lydia
Dorner, Ira Schildkraut and Aneel Aggarwal: “Structure of the
multimodular endonuclease FokI bound to DNA”, Nature, VOL
388, July 1997
- 3. Peter Atkins,
‘Physikalische Chemie’, Wiley-VCH, Third Edition,
January 2002
