2. Model 1: ‘Ready’ – ‘Steady‘- cut
2.1 Reaction kinetics
2.3 Stimulating temperature dependence for a set of parameters
3. Model 2: bind - ‘Ready’- ‘Steady’- cut
3.1 Reaction kinetics
3.3 Different concentrations of Fok_a and Fok_i
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.
each complex is bound to the DNA double
strand, the protein parts can come together to form a heterodimer,
which is the assumption to activate the enzyme.
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:
2.3 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
Fig.6: Higher temperature than optimum, Model 1
Fig.7: Lower temperature than optimum, Model 1
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:
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
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.
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