Onset dynamics of type A botulinum neurotoxin-induced paralysis

Abstract

Experimental studies have demonstrated that botulinum neurotoxin serotype A (BoNT/A) causes flaccid paralysis by a multi-step mechanism. Following its binding to specific receptors at peripheral cholinergic nerve endings, BoNT/A is internalized by receptor-mediated endocytosis. Subsequently its zinc-dependent catalytic domain translocates into the neuroplasm where it cleaves a vesicle-docking protein, SNAP-25, to block neurally evoked cholinergic neurotransmission. We tested the hypothesis that mathematical models having a minimal number of reactions and reactants can simulate published data concerning the onset of paralysis of skeletal muscles induced by BoNT/A at the isolated rat neuromuscular junction (NMJ) and in other systems. Experimental data from several laboratories were simulated with two different models that were represented by sets of coupled, first-order differential equations. In this study, the 3-step sequential model developed by Simpson (J Pharmacol Exp Ther 212:16-21,1980) was used to estimate upper limits of the times during which anti-toxins and other impermeable inhibitors of BoNT/A can exert an effect. The experimentally determined binding reaction rate was verified to be consistent with published estimates for the rate constants for BoNT/A binding to and dissociating from its receptors. Because this 3-step model was not designed to reproduce temporal changes in paralysis with different toxin concentrations, a new BoNT/A species and rate (k(S)) were added at the beginning of the reaction sequence to create a 4-step scheme. This unbound initial species is transformed at a rate determined by k(S) to a free species that is capable of binding. By systematically adjusting the values of k(S), the 4-step model simulated the rapid decline in NMJ function (k(S) >or= 0.01), the less rapid onset of paralysis in mice following i.m. injections (k (S) = 0.001), and the slow onset of the therapeutic effects of BoNT/A (k(S) < 0.001) in man. This minimal modeling approach was not only verified by simulating experimental results, it helped to quantitatively define the time available for an inhibitor to have some effect (t(inhib)) and the relation between this time and the rate of paralysis onset. The 4-step model predicted that as the rate of paralysis becomes slower, the estimated upper limits of (t(inhib)) for impermeable inhibitors become longer. More generally, this modeling approach may be useful in studying the kinetics of other toxins or viruses that invade host cells by similar mechanisms, e.g., receptor-mediated endocytosis.

Fig. 2

Different values of k_L in the 3-step model simulated the frequency-dependence of the rate of onset of BoNT/A-induced paralytic activity. Panel A: plot of data points where values for t₁₀ are expressed as normalized values. Panel B: The 3-step model (dashed line) produced a non-linear relation between k_L (0.01 − 0.3min⁻¹) and the time-to-10% tension, t₁₀. Within the range of experimental data, the plot was linear and at higher values of k_L approached an asymptotic value for the normalized values of t₁₀. Panel C: Values of k_L were adjusted to simulate normalized t₁₀ values at each of the four experimentally used nerve stimulation frequencies. Circles: experimental data from Fig. 4 in [15]; lines in Panels A and C: least-squares fits drawn through the data points

Fig. 1

Simulation of experimental NMJ data with the 3-step model. The time courses of the model species can be compared with the development of BoNT/A-induced paralysis (filled symbols, data from [15]). The time-to-10% tension, t₁₀, was 204 min. The upper limit for the amount of time available for an inhibitor of BoNT/A to exert some effect, tinhib, was ⩽40 min (see Methods). All BoNT/A species are represented as relative values. Dotted line: free BoNT/A; long dashes: bound toxin; short dashes: translocated species; black line: amplitudes of peak twitch tension numerically calculated from the 3-step model; white line (superimposed on black line): amplitudes of peak twitch tension calculated from Eq. 1e

Fig. 4

Time-dependence of the BoNT/A species from the 4-step model that simulated the clinical data. The time courses of the different species can be compared with the relatively slow loss of tension (t₁₀ =15,450 min or 106 days) of BoNT/A-induced paralysis. In contrast to the in vitro NMJ (Fig. 1), the time courses of the unbound BoNT/A and the lytic species overlapped, making the estimated upper limit value of t_inhib longer in vivo than at the isolated NMJ. Closed circles: subjective clinical data from [18]; black line: responses were calculated numerically from the 4-step model using kS = 0.00015 min⁻¹; white line: responses were calculated from Eq. 2c

Fig. 5

Effects of varying k_S on the upper-limit estimates of tinhib and onset times for BoNT/A-induced paralysis. Panel A: the two durations, the times-to-10% peak tension (t₁₀) and the tinhib, increase and converge as the values of k_S decrease. Panel B: the predicted concentration of unbound BoNT/A (decomposed into bulk and free species) at t_inhib, is dependent on the value of k_S. The placement of symbols along the x-axis in both panels (upward triangles, downward triangles, stars, and circles) are defined in Fig. 3 and depict which of the unbound species and values of k_S are associated with those simulations. For clarity, these symbols were placed just above the x-axis to correspond with the appropriate values of k_S

Fig. 3

Different values of k_S in the 4-step model simulated the onset of BoNT/A-induced effects in a variety of in vitro and in vivo data sets. The onset of paralysis with bath-applied BoNT/A at the rat NMJ (circles, from Fig. 1) is rapid and is associated with k_S values ⩾0.01 min⁻¹. In vivo data (time using an exercise wheel, see Methods) from locally injected mice (stars, data from [17]; upward triangles, data from [7]) were simulated using kS =0.001 min⁻¹. Clinical data (subjective measure of relief of neurologic symptoms, see Methods; downward triangles) are from a patient whose first session of BoNT/A treatment was on day 0 (data from [18]). The maximum BoNT/A-induced effect, achieved by day 11, was simulated with k_S =0.00015 min⁻¹