Kinetic Theory of Protein Aggregation Inhibition.

The microscopic mechanisms of irreversible protein aggregation involve a number of steps (Fig. 1A), including primary nucleation, followed by fibril elongation (18). Once a critical quantity of fibrils is formed, however, aggregation is accelerated by secondary nucleation pathways, where the rate of formation of new aggregates depends on the existing aggregate population, leading to exponential growth (19⇓⇓⇓⇓⇓–25); examples of such secondary nucleation pathways include fibril fragmentation (19) and surface-catalyzed secondary nucleation (20⇓⇓⇓⇓–25), which is active in Aβ42 aggregation (22). The combined action of these diverse microscopic aggregation mechanisms on the concentration f(t,j) of aggregates of size j at time t can be quantified via a master equation (SI Appendix, Eq. S4 and subsequent discussion) (14, 16):df(t,j)dt=2k+ Mm(t)f(t,j−1)−2k+ Mm(t)f(t,j) +2k−∑i=j+1∞f(t,i)−k−(j−1)f(t,j) +k1 Mm(t)n1δj,n1+k2 Mm(t)n2δj,n2 ∑i=n2∞if(t,i),[1]where Mm(t) is the monomer concentration; k1,k+,k−,k2 are, respectively, the rate constants for primary nucleation, elongation, fragmentation, and surface-catalyzed secondary nucleation; and n1 and n2 are the reaction orders of the primary and secondary nucleation steps. Summation of Eq. 1 over aggregate size j leads to a set of moment equations (SI Appendix, Eq. S7) for key experimental observables, including the total number concentration of aggregates ca(t)=∑jf(t,j). Solutions to such moment equations lead to a characteristic sigmoidal profile for the aggregate number concentration, with an initial lag phase followed by a saturation phase due to monomer depletion (SI Appendix, Fig. S2). During the initial lag phase, the monomer concentration is approximately constant, Mm(t)≈Mmtot, where Mmtot is the total concentration of monomers. It can be shown (SI Appendix, Eq. S12) that, in this limit, the number concentration of aggregates increases exponentially with time (positive feedback), ca(t)≃(α0/κ0)eκ0t (25), where α0=k1(Mmtot)n1 is the rate of generation of new aggregates through primary nucleation and κ0=2k+Mmtot[k2(Mmtot)n2+k−] is an effective aggregate proliferation rate arising from the combined effect of aggregate growth and multiplication through the secondary nucleation pathways. In the context of inhibiting protein aggregation, a key interest is to block this positive feedback mechanism observed during the early-time exponential growth of aggregates; we will thus focus on the early stages of aggregation (rather than on the saturation phase) and assume a constant concentration for the available soluble monomers throughout. This constant-monomer concentration scenario may also be relevant in vivo, where the monomeric protein concentration is likely to be maintained at constant levels by the action of external mechanisms such as protein synthesis (26).

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Fig. 1.

Elementary molecular events of pathological protein aggregation and the diversity of mechanisms by which a drug can inhibit protein aggregation. (A) Fibrillar aggregates are formed through an initial primary nucleation step followed by elongation. Once a critical concentration of aggregates is reached, secondary nucleation (in the form of fragmentation or, as illustrated in the figure here, surface-catalyzed secondary nucleation) introduces a positive feedback cycle leading to exponential growth of aggregate concentration. (B) A drug can bind monomers; in addition, it can bind primary or secondary oligomers to inhibit primary or surface-catalyzed secondary nucleation. Alternatively, the drug can bind to the fibril ends or the fibril surface to suppress elongation, fragmentation or surface-catalyzed secondary nucleation.

Protein aggregation kinetics can be inhibited in its onset or progression by the presence of a drug through 5 pathways (Fig. 1B) (14): 1) binding to free monomers, 2) binding to oligomers produced by primary nucleation (primary oligomers), 3) binding to oligomers generated by secondary nucleation (secondary oligomers), 4) binding to aggregate ends to block elongation, and 5) binding to the fibril surface to suppress fragmentation or block the production of toxic species through surface-catalyzed secondary nucleation. Since the progression of aggregation is relatively slow compared with the binding rate of drugs, an explicit treatment of the full nonlinear master equation in the presence of a drug shows that, in the limit of constant monomer concentration, the aggregate number concentration ca(t) satisfies (see SI Appendix, section S2 for a derivation)dca(t)dt=α(cd)+κ(cd) ca(t),[2a]where the drug concentration cd affects the rate parameters according toα(cd)=α011+Kmeqcdn111+Kolig,1eqcd,[2b]κ(cd)=κ011+Kmeqcdn2211+Kendseqcd12  11+Ksurfeqcd1211+Kolig,2eqcd12.[2c]Note that the kinetic equation for aggregate concentration and the drug-dependent rate parameters (Eq. 2) can be explicitly derived from a microscopic description of aggregation inhibition through a nonlinear master equation describing the time evolution of the entire aggregate size distribution. They provide a link between microscopic mechanisms of aggregation and inhibition to macroscopic aggregation measurements. The complex interplay between the multiple aggregation pathways and the drug is captured explicitly by renormalized kinetic parameters α(cd) and κ(cd), which depend on the drug concentration cd and are specific functions of the kinetic parameters of aggregation as well as the equilibrium binding constant of the drug to the targeted species, K×eq. Here, × is a placeholder for the target species and the respective pathway, i.e., monomers (m), primary or secondary oligomers (olig,1 and olig,2), fibril ends (ends), and fibril surface sites (surf). In Eq. 2, we have focused on the total aggregate particle concentration; it has, however, been shown that low molecular weight oligomers are key cytotoxic species linked to protein aggregation (27⇓–29). To account for this situation, in SI Appendix, Eq. S15, we show that, in the constant-monomer concentration limit, a linear proportionality relationship links ca(t) to the concentration of oligomers. Thus, after appropriate rescaling of concentration, the same Eq. 2 can be used to describe oligomeric populations as well. Throughout this paper, we thus use the generic term “aggregate” to refer to the relevant population of toxic aggregate species.

Optimal Control of Protein Aggregation.

To find the optimal therapeutic treatment that inhibits the formation of toxic aggregate species requires a cost functional that balances aggregate toxicity against drug toxicity:C=Costca(t),cd(t)=∫0Tdt Lca(t),cd(t) ,[3]where T is the total available time for treatment, and L is a function that characterizes the cost rate that increases for larger aggregate and drug concentrations. L is expected to be a nonlinear and monotonically increasing function of drug and aggregate concentrations. In the absence of detailed experimental insights into the form of L, we linearize and use the following monotonous form of the cost function L=ca(t)+ζ cd(t), where ζ>0 quantifies the relative toxicity of aggregate and drug molecules. In SI Appendix, section S3.7, we show that the predictions from the linearized cost function remain qualitatively valid also in the case of a nonlinear cost function of the form L=ca(t)n+ζ cd(t)n. Future experiments may provide detailed insights into the specific form of the cost function allowing to scrutinize our prediction quantitatively. The optimal drug administration protocol cd(t) minimizes the cost functional Eq. 3 given the aggregation dynamics governed by Eq. 2, thus enabling us to couch our problem within the realm of classical optimal control theory (17) that allows for bang-bang control solutions, given the linear nature of the cost function.

Indeed, the optimal treatment protocol consists of using piece-wise constant concentration levels of the drug over varying time spans of the treatment (Fig. 2 A and B) determined by the drug toxicity, the aggregation kinetic parameters, and the mechanism of inhibition (SI Appendix, section S3). In this protocol, T1 is the waiting time for drug administration, T2−T1, denotes the time period during which the drug is applied, and T−T2, is a drug-free period after treatment. We find that, depending on whether the drug suppresses selectively primary nucleation or secondary nucleation and growth at the ends of the aggregates, the optimal protocol for drug administration is fundamentally distinct (30).^† When the drug inhibits primary nucleation (α=α(cd), κ=κ0; Fig. 2A), there is no waiting period for drug administration (T1=0, “early administration”), and the optimal treatment duration readsT2=T−1κ0lnζcdκ0α0−α.[4]When the drug affects secondary nucleation or elongation (κ=κ(cd), α=α0; Fig. 2B), the optimal protocol is qualitatively different: the drug must be administered after a waiting period T1 (“late administration”) and the optimal treatment duration isT2−T1=κ0κ0−κT−1κ0lnζcdκ02α0(κ0−κ).[5]In either case, the optimal treatment time decreases with increasing drug concentration or toxicity. Moreover, at low drug concentrations, there is a regime where the drug must be administered for the full time period T, while if the drug concentration exceeds a critical threshold, cd>(α0/ζκ0) eκ0T, the preferable choice is no treatment. The optimal treatment duration corresponds to a minimum in cost and reflects the competition between drug-induced suppression of aggregates and drug toxicity (Fig. 3A). The achievability of optimal treatment conditions is determined by the curvature of the cost function at the optimal treatment, which approximately reads (κ0−κ)ζcd (SI Appendix, section S3.4.4). Lower curvatures around the optimal treatment parameters facilitates a robust possibility to find mostly optimal treatment conditions.

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Fig. 2.

Distinct optimal treatment protocols characterize the timing of drug administration for compounds that inhibit primary or secondary nucleation processes. (A) Optimal treatment protocol for the administration of a drug that inhibits primary nucleation (Top). In this case, the drug must be administered as early as possible (T1=0) and for a duration T2. Increasing drug concentration decreases the overall duration T2 of the optimal treatment (Bottom) but without affecting the need for an early administration. When the drug concentration is large, no treatment is favorable (pink), while at low drug concentrations, the optimal treatment can take the full available time T (green). (B) For a drug that inhibits either fibril elongation or secondary nucleation, a late, rather than early, administration of the drug is required (Top). The optimal treatment protocol is thus characterized by 2 switching times, T1 and T2, that define the start and the end of drug administration, respectively (Bottom). The duration of the treatment, T2−T1, decreases with increasing concentration of the drug. The parameters used in the plots are: ζκ0/(α0Koligo,1eq)=0.6, κ0T=1.3 (A); and ζκ0/(α0Ksurfeq)=10, κ0T=4.5 (B).

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Fig. 3.

Comparison between different inhibition strategies and predictions of lifetime gain due to optimal treatment. (A) The normalized cost, Cost/Cost(cd=0), for the inhibition of secondary nucleation, has a minimum (Eq. 5) as function of the dimensionless treatment duration κ0(T2−T1). At lower drug concentration (green line), the minimum of the cost becomes broader, indicating an easier access to the optimal protocol in the presence of fluctuations or limited knowledge of cellular kinetic parameters or concentrations. (B) Phase diagram indicating the region of parameter space where inhibition of primary nucleation has a lower cost than inhibition of secondary nucleation or growth. The green line indicates how the boundary line shifts when drug toxicity is increased by a factor of 2. Similarly, the pink line indicates how the boundary line shifts when the system is preaged, i.e., has an increased concentration of aggregates initially. Note that cdκ/α≃(κ0/α0)cd3/2K1oeq/K2ndeq, where K1oeq and K2ndeq are the binding constants (affinities) for the inhibition, respectively, of primary and secondary nucleation. Thus, decreasing K1oeq or increasing K2ndeq favors the inhibition of secondary nucleation over primary nucleation. (C) Cost without drug (blue) and optimal cost (pink) as a function of available time κ0T. Note the dramatic difference in the time dependence of the cost for the optimal treatment (linear in T) and without treatment (exponential in T). (D) Expected life expectancy as a function of treatment duration. There is a distinct maximum where the gain in life time is maximal in correspondence of the optimal treatment protocol. The parameters used in the plots are α0/κ0=2×10−8, ζ=200, Ksurfeq=5 μM−1, κ0T=13, cd=2 μM (green), cd=6 μM (blue) (A); and κ0Costc=10−3.5 M, α0/κ0=10−7, ζ=10, Ksurfeq=1 μM−1, cd=3 μM (green), cd=5 μM (blue) (D).

Our optimization approach allows to use the cost function to compare quantitatively different inhibition strategies and to identify the regions in the parameter space where a certain strategy is to be preferred over an other; we illustrate this idea by comparing the costs for inhibition of primary or secondary nucleation (Fig. 3B and SI Appendix, section S3.4.6). We find that at large drug concentrations, and short available times κ0T, the inhibition of primary nucleation represents the optimal treatment strategy compared with the inhibition of secondary nucleation or elongation, as the former strategy exhibits lower cost. Indeed, a drug that inhibits primary nucleation must be administered from the beginning. Hence, preventing aggregation over a longer time κ0T necessarily requires longer periods of drug administration, eventually making the inhibition of primary nucleation costlier than blocking secondary nucleation at later stages. A boundary line, corresponding to equal costs for both strategies, separates the regimes of optimal treatment. The position of the boundary line depends on the relative affinity of the drug to the primary oligomers compared with secondary oligomers, fibril ends, or fibril surfaces. Increasing drug toxicity shifts the boundary line to the right, hence favoring inhibition of primary nucleation. Another interesting parameter to consider is the initial level of aggregates, which provides a measure of preaging of the system; we find (SI Appendix, section S3.6) that increasing the initial level of aggregates shifts the boundary line to the left, hence disfavoring the inhibition of primary nucleation. Overall, for known values of the relative toxicity, our approach suggests how to select specific drugs corresponding to different mechanisms of action either in an early or late stage of the detection of protein aggregation disorders and depending on experimentally accessible parameters, such as drug affinity.

We next use the cost function to characterize longevity gain as a function of the parameters of drug-induced inhibition of aggregation (Fig. 3C and SI Appendix, section S3.4.5). We define the life time as the time at which the cost reaches a critical value corresponding to the cost that a cell or an organism can tolerate before it dies. In the absence of any drug treatment, the cost function grows exponentially with available time T, i.e., Cost(cd=0)≃(α0/κ02)eκ0T. Crucially, the addition of a drug following the optimized treatment protocol lowers the cost down to a linear increase in time, Costopt≃ζcdT. Hence, the difference in life times between an optimized treatment and the situation when no treatment is applied can be significant. The expected life time as a function of treatment duration displays a distinct maximum where the gain in longevity is maximal in correspondence of the optimal treatment protocol (Fig. 3D). The maximal life expectancy decreases with increasing drug concentration.

Comparison with Experiments.

We finally tested qualitatively the efficacy of the optimal protocol in practice by considering previous data (10, 15) on the inhibition of Aβ42 amyloid fibril formation of Alzheimer’s disease using the drug Bexarotene in a C. elegans model of Aβ42-induced dysfunction (Fig. 4A) (10, 31). Fig. 4B shows the effect of administering increasing concentrations of Bexarotene to Aβ42 worms in their larval stages on the frequency of body bends, a key parameter that indicates the viability of worms. At low drug concentrations, increasing Bexarotene concentration has beneficial effects on worm fitness, but too large drug concentrations decrease worm fitness. Thus, there is an optimal dose of Bexarotene (10 μM) that leads to maximal the recovery of the worms. This optimal dose emerges from the competition between the inhibition of protein aggregation by Bexarotene (SI Appendix, Fig. S6 A and B) and its toxicity (SI Appendix, Fig. S6C), as anticipated by our cost function (SI Appendix, section S3.5). At a mechanistic level, Bexarotene has been shown to affect protein aggregation by inhibiting selectively primary oligomers and hence reduce primary nucleation both in vitro (10) and in the C. elegans model of Aβ42-induced toxicity (10) (SI Appendix, Fig. S6A). Thus, the key prediction from our model is that Bexarotene would be most effective with an early administration protocol. This prediction is in line with the experimental observations (Fig. 4C) (10) that show that the administration of Bexarotene following a late administration protocol at day 2 of worm adulthood does not induce any observable improvement in fitness relative to untreated worms. In contrast, administering Bexarotene at the onset of the disease in the larval stages (early administration), leads to a significant recovery of worm mobility. To further support our predictions, we consider in Fig. 4D the inhibition of Aβ42 aggregation by another compound, DesAb₂₉₋₃₆, which has previously been shown to inhibit selectively secondary nucleation (15). The data in this case show that DesAb₂₉₋₃₆ is more efficacious when administered at late times than during the early stages of aggregation, an observation that is in line with the theoretical predictions of our model.

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Fig. 4.

Application to the inhibition of Alzheimer’s Aβ42 aggregation in C. elegans model of Aβ42-mediated toxicity. (A) Expression of Aβ42 in the worm’s muscle cells leads to age-progressive paralysis detected through the reduction in the frequency of body bends relative to healthy worms, which do not express Aβ42. (B) Low drug (Bexarotene) concentration, which selectively inhibits primary nucleation, improves worm fitness due to the inhibition of protein aggregation; however, too large drug concentrations decrease worm fitness due to toxicity of the drug (data from ref. 10). (C) Effect of early (72 h before day 0 of adulthood) and late (day 2 of adulthood) administration of Bexarotene (10 μM) show that early administration is significantly more effective in alleviating the symptoms of worm paralysis compared with the late administration of the same drug. In the latter case, there was no observable improvement of worm fitness compared with untreated Aβ42 worms (data from ref. 10). (D) Effect of early (day 1 of adulthood) and late (day 6 of adulthood) administration of a selective inhibitor of secondary nucleation (DesAb₂₉₋₃₆) (data from ref. 15) show that a late administration of DesAb₂₉₋₃₆ is more effective than an early administration in causing worm recovery. In B and C, the effect on fitness (body bends per second) was measured at day 6 of adulthood and compared with healthy worms and untreated Aβ42 worms, while in D, the effect on fitness was measured at day 7 of adulthood. Error bars indicate the SEM; the sample size was n=200 worms for Bexarotene experiments (10) and n=500 worms for DesAb₂₉₋₃₆ experiments (15).