Team:IISc-Bangalore/Model
Model
SpyTag - SpyDock
The SpyTag/SpyCatcher system is a peptide-protein system that provides a simple and reliable method of binding our organophosphate hydrolases of interest to the bacterial cellulose sheets, due to its near infinite affinity and rapid irreversible isopeptide bond formation. The most recent iteration of the system, SpyTag003:SpyCatcher003, surpasses its predecessors by a significant margin, with a bimolecular rate constant of \( 10^{5} - 10^{6} \ M^{-1} s^{-1} \)
One of our aims for this project was to achieve a plug-and-play system within our filter. The idea was to design the functionalized BC sheet in such a manner to enable switching out of the enzymes. However, the formation of the isopeptide bond for SpyTag/SpyCatcher is irreversible, making enzyme switching not feasible.
The SpyTag/SpyDock system is a similar protein-peptide system with a modified SpyCatcher protein which prevents the isopeptide bond formation, keeping the interaction noncovalent. This looked like a promising option since the noncovalent interaction showed minimal leak-through during washes, but modelling the SpyTag/SpyDock system showed that for even a moderate rate of flow, the protein-peptide complex is dissociated in a matter of minutes.
We design a first-order model to observe this. A more detailed analysis involving shearing rate would be more accurate, but the prediction made by the first order model is enough to rule out SpyDock as a viable option.
-
SpyDock binds with SpyTag002 reversibly with a bimolecular rate constant of \( k = (2.0 \pm 0.2) \times 10^{-4} M^{-1} s^{-1} \)
-
The dissociation constant is \( K_d = (0.073 \pm 0.013) \times 10^{-6} M \), as determined by isothermal titration calorimetry for SpyTag-MBP/SpyDock.
-
The BC sheet will be present in the form of a film on a microchannel with radius \(r\) and the fluid to be purified will be passing through with volumetric flow rate \(Q\).
Assuming that the reversible reaction is of the form, where \([A]\) is the SpyDock substrate and \([B]\) is the SpyTag002-protein complex
\[ [AB] \rightleftharpoons [A] + [B] \]The differential equations are:
\[ \frac{d[AB]}{dt} = k[A][B] - K_d k [AB] \] \[ \frac{d[A]}{dt} = K_d k [AB] - k[A][B] \] \[ \frac{d[B]}{dt} = K_d k [AB] - k[A][B] - \alpha Q [B] \]The proportionality constant \( \alpha \) determines the rate at which \(B\) gets washed out in the fluid. We expect it to be directly proportional to the flow rate and the concentration of SpyTag.
We model these differential equations in Python with estimates of \(\alpha\) and \(Q\). The results are plotted below:
We see that the SpyTag/SpyDock interaction gets abolished and the non-covalently associated protein complex gets washed away for all cases. Our model has thus safely ruled out the use of SpyDock/SpyCatcher as a reliable method for implementation of the functionalized BC sheet.
OP detection and minimum concentration
The enzyme kinetics of organophosphate hydrolase OpdA has been well quantified by multiple studies, with the reactions following Michaelis-Menten kinetics.
The design for our filter involves a polydimethylsiloxane (PDMS) microchannel coated with the functionalized BC film, with the organophosphate-laden fluid passing through the channel. For implementation of the filter, a rough estimate of the fluid flow rate as well as the required length of the microchannel is needed. This lets us optimize filtration by adjusting the flow rate appropriately.
We design a basic model from first principles to determine the rate of degradation of organophosphates by the filter. We design our model based on the following assumptions:
-
The fluid in the microchannel moves with a velocity of \(v_f\) through the tube.
-
The substrate concentration is in steady state, i.e., it remains unchanged in time.
-
Due to the small radius of the microchannel \( \sim 10 \mu m \), there is no appreciable change in OP concentration with radius.
-
Not all the fluid in the microchannel can react with the enzymes on the surface. Therefore, we characterize the size of reacting region with a dimensionless parameter \( \eta \) (see figure below). This parameter is considered to be small ( \(\eta \lt 1 \)).
-
From this, we conclude that the OP concentration \([S]\) only changes along the length of the tube.
We consider a segment of our microchannel tube at a distance \(x\) from the inlet, of length \(dx\).
We assume that the region between the cylinders is where the reaction occurs. For small \(\eta\),
\[ \Delta V \approx 2 \pi r (\eta r) dx = 2 \eta \pi r^2 dx \]The ratio of the reacting volume to the total volume is
\[ \frac{\Delta V}{V} = 2 \eta \]Since the hydrolysis is assumed to follow Michaelis-Menten kinetics, the rate of degradation of the organophosphate is given by:
\[ k = \frac{d[S]}{dt} = -k_{cat} [E_0] \frac{[S]}{K_M + [S]} \]You can find the definitions of all the variables here.
This is the state of the fluid in the element at time \(t\). In time t+dt, fluid from this region rushes into the region \(x+dx\). The relation between \(dt\) and \(dx\) is given by:
\[ dx = v_f dt \]In this time, the substrate concentration in the fluid element has reduced since the fluid in the outer region has been degraded by enzyme activity. The new concentration is given by:
\[ [S]_{new} = \frac{(V-\Delta V)[S](x) + \Delta V \left(\frac{d[S]}{dt}\right) dt}{V} \] \[ = (1 - 2\eta)[S](x) - 2 \eta k_{cat} [E_0] \frac{[S]}{K_M + [S]} \]But since the fluid must be in steady state, the new concentration must equal the concentration of the fluid previously present at that location, i.e.,
\[ [S](x + dx) = [S]_{new} \]Substituting our expression and replacing \( dt = \frac{dx}{v_f} \), we get the differential equation describing the concentration gradient in the microchannel.
\[ \frac{d[S]}{dx} = - \frac{2 \eta k_{cat}}{v_f} [E_0] \frac{[S]}{K_M + [S]} \]This equation agrees with intuition - the faster the flow, the smaller the gradient \(\frac{d[S]}{dx}\), which means that the organophosphate hardly gets degraded as it passes through the tube. Similarly, larger the effective volume of cross section \((\eta\)), larger the gradient (see figure below).
To find the gradient for a typical filtration process, we estimate the values of the constants above from literature (here).
Once our filter is implemented, these models will be fit on the data obtained to estimate \(\eta\) and the initial enzyme concentration \([E_0]\). Implementing these fits will enable us to optimise the flow rate and the microchannel length for efficient filtration of organophosphates.
References
-
Jackson et al. (2005). The effects of substrate orientation on the mechanism of a phosphotriesterase. Organic & Biomolecular Chemistry, 3(24), 4343
-
Anuar et al. (2019). Spy&Go purification of SpyTag-proteins using pseudo-SpyCatcher to access an oligomerization toolbox. Nature Communications, 10(1), 1734
-
Keeble et al. (2019). Approaching infinite affinity through engineering of peptide - protein interaction. Proceedings of the National Academy of Sciences, 116(52), 26523-26533