Team:Freiburg/Modeling/Parameter analysis

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Parameter estimations for the 2 pools TCR dimerization model

Evolving and regarding the equations was the first step to begin to understand how the time course of the involved quantities could behave. To get a deeper insight into the characteristics of the model, the behaviour for different parameters and parameter sets is analyzed. Starting with all parameters at value one except one, even though not very realistic, gives a first clue about the course in time of the active TCR density, the interface pool density and the spare pool density.

Figure 1: TCR densities in time

chosen parameters: s = 1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 1;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 1;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 1;  % NIP amount l = 1;  % ratio p = 1;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

Different parameter set

The turnover rate (S) is set to a value 10 times smaller than the previous value and the amount of the NIP is now 4 times smaller than in the calculation above. The TCR-NIP dissociating rate (k_off) is 5 times smaller than the binding rate (k_on). The same holds for the dimer dissociating rate (k_doff) in relation to the dimerization rate (k_don).

Figure 2: TCR densities in time for a chosen set of parameters

chosen parameters: s = 0.1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 0.2;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 0.2;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 0.25;  % NIP amount l = 1;  % ratio p = 1;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

The difference in values between binding rate and dissociating rate, respectively between dimerization rate and dimer dissociationg rate leads to a higher TCR activity then in the calculation above where k_on=k_off=k_don=k_doff=1, due to the formation of more monomeric TCR-NIP complexes and TCR-NIP dimers.
In generall an increase in k_on or k_don causes a TCR activity elevation. An increase in k_off or k_doff gives a faster TCR-NIP complex and TCR-NIP dimer dissociation, so a decrease in TCR activity.

TCR activity for different binding rates and dimerization rates

In the calculations for the plots below the same parameters as above were chosen except for k_on repectively k_don. The black lines represents active TCR densities in time for different k_on repectively k_don.

Figure 3.1: TCR activity in time for different binding rates (kon)

Figure 3.2: TCR activity in time for different dimerization rates (kdon)

Increasing the binding rate gives a little higher active TCR densities than increasing the dimerization rate in the same way.

TCR activity dependent on NIP amount

The next two plots show how the TCR activity in time differs when different amounts of NIP are used:

low NIP amount: Figure 4: Active TCR density in time with low NIP Amount (5% of initial TCR in membrane)

chosen parameters: s = 0.1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 0.2;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 0.2;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 0.1;  % NIP amount l = 1;  % ratio p = 1;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

The TCR activity is low when the NIP amount or concentration is low. With an unsufficient amount of NIP many TCRs remain unbound, thus dimerization is low, so the activity remains lower, too.

high NIP amount: Figure 5: Active TCR density in time with high NIP amount (30% of initial TCR in membrane)

chosen parameters: s = 0.1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 0.2;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 0.2;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 0.6;  % NIP amount l = 1;  % ratio p = 1;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

An addition of a high NIP amount leads to high TCR activity, as more TCR-NIP complexes are build and hence more dimers.

TCR activity for different amount of NIP (parameters as in the previous plot):

Figure 6: Active TCR density in time with different NIP amounts

The more NIP is used, the higher is the activity.

TCR activity dependent on exchange between spare and interface pool

The TCR can switch from a non-binding state to a binding state or in other words, the TCRs in the spare pool can become TCRs in the interface pool. This exchange is regulated by the parameters λ and φ. λ is a ratio between the spare and the interface pool and φ is the exchange rate constant.

low exchange: Figure 7: Active TCR density in time (low exchange: λ=5 and φ=0.05)

chosen parameters: s = 0.1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 0.2;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 0.2;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 0.25;  % NIP amount l = 5;  % ratio p = 0.05;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

For a very high ratio λ and a low exchange rate φ the ratio of the spare pool (non-binding TCR) to the interface pool (binding TCR) is high and the switching of a TCR between the non-binding and the binding state is low. Thus there is a low TCR activity because not enough TCRs are accessible by NIP. This can be due to TCRs which are surrounded by big sized proteins who avoid a TCR-NIP formation.

high exchange: Figure 8: Active TCR density in time (high exchange: λ=0.2 and φ=5)

chosen parameters: s = 0.1;  % turnover rate kon = 1;  % TCR-NIP binding rate koff = 0.2;  % TCR-NIP dissociating rate kdon = 1;  % TCR-NIP dimerization rate kdoff = 0.2;  % dimer dissociating rate ka = 1;  % activation rate ki = 1;  % internalization rate N = 0.25;  % NIP amount l = 0.2;  % ratio p = 5;  % exchange rate h = 2;  % kinetic order initial integration conditions: S0 = 1;  % spare TCR density; T0 = 1;  % interface TCR density; TA0 = 0;  % active TCR density;

A low ratio λ and a high exchange rate φ lead to a high TCR activity as enough TCRs are available in the interface pool for NIP binding. The fast changing from the non-binding to the binding state assures a sufficient amount of TCRs for TCR-NIP and then dimer formation.

TCR activity for different exchanges (parameters as in the plot above except: N=0.2)

active TCR density in time for different exchanges

The x-axis represents the time course of the activity, the y-axis represents both parameters φ ( = y) and λ ( = 2 - y). So each black line in the plot is a time course of the TCR activity for a different φ and λ. The z-axis is the response intensity. With increasing exchange between the interface and spare pool, more TCRs switch to the binding state, hence more TCRs can bind NIP. As a consequence the active TCR density is higher then for a low exchange.