Team:Montpellier/Model

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MODELLING

Our intuition tells that KARMA (protease linked to an antibody) improves efficiency and specificity of a protease alone. In order to formalise that, we developed a model based on a system of Ordinary Differential Equations (ODEs) that is capable to follow the kinetics of the molecular complexes involved in the process. Our results actually confirm our hypothesis that KARMA could theoretically improve proteolysis efficiency.

Below we explain how we developed the model, starting from the popular Michaelis-Menten (MM) model that describes the kinetics of enzymatic reactions and that we use as a reference for studying the functioning of a protease alone.

Michaelis-Menten kinetics as a model for proteolysis kinetics

This model was proposed by Henri [1] to describe a single enzyme-substrate (E-S) reactions and has been given experimental evidence later by Michaelis and Menten [2].

This model considers an enzyme E that binds to a substrate S to form a complex ES.
This complex then releases a product P, and the original enzyme is regenerated. This may be represented schematically as

For our problem, the enzyme E is a protease targeting the substrate protein S, and the product P is the proteolysis of this target.

The reactions above can be described by a system of four nonlinear ODEs representing the rate of change of the reactants in time:

Under certain assumptions, e.g. quasi-steady-state condition in which the concentration [ES] does not change on the timescale of product formation, one obtains:

where K_M is the Michaelis constant. The Michaelis constant is numerically equal to [S] when the reaction rate is at half-maximum, and it characterises the affinity of an enzyme for its substrate. This means that enzymes with high values of K_M require high substrate concentration to produce an important enzymatic activity, which can be expressed as a weak affinity and vice-versa. We remind that:

KARMA kinetics

The Principle of KARMA

We now extend this model to study a protease linked to an antibody, which targets the particular protein that we want to degrade. Once the antibody specifically binds its target, the concentration of the tethered proteases nearby the target’s cleaving site effectively increases. To give an idea of the variation of the concentration of the protease around the cleaving site, keep in mind that a single copy protease in a bacterium has a concentration of ~1 nM, but once the antibody binds the target, its concentration around the cleaving site increases to ~10² M, i.e. KARMA leads to 10¹¹ fold increase in the effective concentration of the protease once the antibody has bound.

Figure 1: Scheme showing the local volume V_K=4πL³/3 delimited by KARMA. The presence of a substrate in such a confined volume yields a concentration C_K=1/(V_KN_A), where N_A is the Avogradro’s number.

This is the rationale behind KARMA. Now we need to couple the antibody on/off kinetics on the target and the protease kinetics, previously introduced with the MM model.

Cooperativity between antibody and protease

KARMA can act in two distinct ways: either the protease binds the cleaving site first, or the antibody first binds the target and highly increases the concentration of the protease around the cleaving site. In the former situation KARMA behaves as a standard protease, and it follows a MM kinetics -top branch in Figure 2-. This is a non-specific pathway, determined by the characteristics of the protease alone. The latter -bottom branch- is the characteristic pathway of KARMA, specific to the antibody target. This problem can be seen as an extension of the MM model with a random ordered multiple binding.

Figure 2: The two pathways of KARMA. The top one represents the kinetics of the protease alone (i.e. non-specific). The bottom one is KARMA’s characteristic pathways (specific) driven by the early attachment of the antibody to the target protein.

We also noticed that, after proteolysis, the antibody could stay attached to its target site (final product on the right on Figure 2), and then KARMA would act as a standard protease. This is why KARMA, as we will see, works efficiently at short timescales. More in detail, the product of proteolysis (see Figure 2) follows additional recycling pathways as shown in the scheme below (Figure 3). We considered these additional reactions in our model and we discuss antibody recycling later on.

Figure 3: KARMA remains attached to a degraded protein until the antibody unbinds (upper branch) or behaves as a simple protease in the meantime (bottom branch).

We thus defined a system of ODEs describing the time evolution of the molecular complexes involved in the process:

We summarise in Table 1 the symbols used for all reactants involved in the process, and their description. In Table 2 we list the rate constants and the values used in our simulations.

Table 1: Different symbols used in the model and their corresponding variables in the system of ODEs.

Table 2: Parameters found in the literature for the VHH (antibody) and TEV protease used in our experimental proof of concept.

KARMA improves the efficiency of a protease

We simulated our system taking into consideration parameters found in the literature (see Table 2). We compare the protein degradation carried out by a single protease —following Eqs. (2)-(5)— and KARMA —following Eqs. (8)-(18)—, and the results are plotted in Figure 4. In this figure we show the percentage of protein targets as a function of time. KARMA, in black, acts faster compared to the protease alone, in yellow. This demonstrates that the concept of KARMA can actually improve the efficiency of proteases.

Figure 4: Percentages of targeted proteins over time for different Protein/Enzyme ratios.

Moreover, in the previous plots we highlight the contribution of KARMA to proteolysis from its specific pathway (dashed black curve) and the non-specific pathways (dotted line).

KARMA as a specific degradation tool

One of the objectives of our project is to make a specific molecular tool, minimising off-target proteolysis. The aim is to increase the specific pathway at the expenses of the standard non-specific one of the protease alone.

We thus changed the affinity of the protease in order to make the non-specific pathway ineffective, and checked if KARMA, thanks to its specific pathway, can still efficiently degrade the target. To do that we have strongly reduced the kon of the protease (90% and 99% reduction). The results are shown in Figure 5.

Figure 5 : an inefficient protease substantially does not affect the protein concentration, while KARMA still shows a degradation effect. WT indicates the wild-type protease (alone and in KARMA), while mut are mutant proteases with their kon re duced by 90% and 99%.

Multi-substrate simulations

Now that we have understood how we can enhance KARMA’s specificity at the expenses of the protease’s efficiency, we can test the model on a more realistic situation in which the targeted protein is mixed with many other off-target proteins.

Figure 6: A substrate for which KARMA’s antibody is specific (right) and one for which it is not (left).

The challenge is to efficiently degrade the target protein without degrading other substrates. We have hence added a second substrate to our system of equations, which is not recognised by KARMA’s antibody but that could be degraded by the protease. We then performed a simulation with the mutant protease with reduced affinity (-99% mut, see above) to reduce the non-specific pathway of KARMA.

The simulations show that KARMA mainly degrades the antibody’s substrate, and leaves the other substrate mainly unaffected, thus obtaining the desired effect.

Figure 7: Percentages of targeted (specific substrate) and non targeted (non-specific substrate - other) proteins as a function of time. We positioned ourselves in realistic conditions in which the background substrate is in excess ( [Other proteins]₀ = 1000 × [Targeted protein]₀ ).

Conclusions

To summarise, our model shows that:
• KARMA acts faster and more efficiently than a protease alone.
• We can reduce the efficiency of a protease to avoid the non-specific pathway and make KARMA specific for a given target.
• When in contact with many substrates, KARMA acts mainly on the target molecules.

Our model thus shows that KARMA can turn a non-specific and low efficient protease into a specific and more efficient proteolytic tool.

Perspective: Antibody segregation and recycling

However, we realised that, due to the strength of the antibody binding ( K_offV ), we expect that KARMA stays attached to the cleaved molecule for a long time. It would then act like a simple protease that tows a degraded protein in its search for a new one to cleave.

KARMA then works efficiently at short times, before the saturation of the complex with the antibody bound to the target. After that, the kinetics is driven by the detachment of the antibody, i.e. the recycling capability of KARMA.

Only from time to time an antibody detaching from its previous target permits the conventional processing of KARMA as a specific tool. Similarly, an antibody binding to already degraded proteins may inhibit the normal functioning of our molecule.

Our modelling then informs that, as a perspective, KARMA’s functioning can be optimised by a design that facilitates recycling and antibody detachment after proteolysis.

METHODS

"A good ODE integrator should exert some adaptive control over its own progress, making frequent changes in its step size. Usually, the purpose of this adaptive stepsize control is to achieve some predetermined accuracy in the solution with minimum computational effort. Many small steps should tiptoe through treacherous terrain, while a few great strides should speed through the smooth uninteresting countryside. The resulting gains in efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten, a hundred, or more. Sometimes accuracy may be demanded not directly in the solution itself, but in some related conserved quantity that can be monitored."[6]

Numerical integration is a powerful tool that becomes fundamentally useful when we want to resolve equations that govern a system for which we don't know any analytical solution (apart of the steady state). This is even more true if the system that we study has few symmetry properties and a lot of degrees of freedom or highly coupled variables like it's the case for KARMA's kinetics. To overcome this intricacy without having to assume tricky approximations, we have chosen to use a numerical integration method with an adaptive stepsize, and one of the best techniques of the kind is the “Adaptive 4th order Runge-Kutta (A-RK4)". A 4th order method, means that the total accumulated error is on the order of Oh⁴.

the Runge–Kutta methods are a family of implicit and explicit iterative methods used to numerically solve differential equation. The commonly used Euler method corresponds to the 1st order RK method. Although, it is very easy to implement it remains limited in scope due to its sloppy behaviour and large truncation error.

Following this link you can download a larger and more exhaustive method of the model analysis :

HERE

References

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[1] V. Henri (1902) Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris 135,916–919

[2] https://fr.wikipedia.org/wiki/%C3%89quation_de_Michaelis-Menten

[3] http://www.med-tohoku-antibody.com/file/reference/PEP2018_PAWtag.pdf

[4] https://pdfs.semanticscholar.org/9da4/a4f810da036d3d098329e62ce617b49c43e9.pdf

[5] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3009406/

[6] http://perso.fundp.ac.be/~amayer/Cours/ApprocheNumerique/chap16.pdf