Neurodegeneration in Huntington disease is described by neuronal loss in which the probability of cell death remains constant with time1. However, the quantitative connection between the kinetics of cell death and the molecular mechanism initiating neurodegeneration remains unclear. One hypothesis is that nucleation of protein aggregates containing exon I fragments of the mutant huntingtin protein (mhttex1), which contains an expanded polyglutamine region in patients with the disease, is the explanation for the infrequent but steady occurrence of neuronal death, resulting in adult onset of the disease2. Recent in vitro evidence suggests that sufficiently long polyglutamine peptides undergo a unimolecular conformational change to form a nucleus that seeds aggregation3. Here we use this nucleation mechanism as the basis to derive a stochastic mathematical model describing the probability of aggregate formation in cells as a function of time and mhttex1 protein concentration, and validate the model experimentally. These findings suggest that therapeutic strategies for Huntington disease predicated on reducing the rate of mhttex1 aggregation need only make modest reductions in huntingtin expression level to substantially increase the delay time until aggregate formation.
Huntington disease is an autosomal dominant neurodegenerative disorder resulting from the expansion of a CAG repeat region in the gene encoding huntingtin (htt). Intracellular aggregates consisting of an N-terminal fragment of mutant htt are observed in postmortem tissue4; however, the role of these protein aggregates in the pathogenesis of the disease is controversial5, 6, 7, 8, 9, 10. Many adult-onset neurodegenerative diseases, including Huntington disease, involve a first-order exponential decline in surviving neurons with time1. This, combined with the late onset and slow progression of these diseases, implies that very rare single-hit events rather than cumulative damage underlie the neurodegenerative process. We consider here alternative models linking the molecular mechanism of aggregate nucleation to the stochastic formation of visible aggregates within cells.
Modeling of chemical reactions on the test-tube scale makes use of the approximation that the system is continuous and homogeneous and therefore may be treated as deterministic3, 11. This approximation works well when the system contains a large number of molecules, because discrete events are averaged out. In a cellular context the concentrations of most proteins are low and the total volume is small, so there are perhaps only a few thousand copies of many proteins. In this context, it is often more appropriate to treat reactions as discrete, stochastic events, as done in this work12.
The mechanism of polyglutamine peptide aggregation in vitro involves a misfolded monomeric nucleus, assumed to be in equilibrium with its properly folded counterpart3. Nucleated polymerization kinetics are exemplified by long lag times followed by rapid aggregate growth, with a strong dependence of aggregation lag time on monomer concentration; the observed lag time for gelation of sickle-cell deoxyhemoglobin in vitro decreases proportionally to concentration to the 30th power13. Nucleation-dependent polymerization has been proposed to govern disease progression kinetics in Alzheimer and prion-related diseases14.
In the nucleation model considered here (Fig. 1), the formation of a stable aggregate nucleus is assumed to be an irreversible step that leads to visible aggregate formation following a growth time that is shorter than the time interval between successive fluorescence microscopy images in the cell culture experiments described here (15 min) and much shorter than the overall time scale of the experiment (18 h). The abruptness of visible aggregate formation is an empirical result; there is negligible ambiguity in scoring cells showing aggregates (for example, Fig. 2). The time of aggregate appearance is taken as a proxy for the time of nucleus formation, a workable assumption if the growth time between nucleation and visible aggregate appearance is very short compared with the lag time for nucleus formation. It is unclear technically how to detect the rare event of individual nucleus formation in a single cell in order to directly measure visible aggregate growth time; in this work we tested the assumption only by consistency of model predictions with measured rates of visible aggregate formation.
Figure 1: Hypothesized mechanism of httex1 aggregate formation3 (Models 1, 1a and 1b in Supplementary Methods).
Httex1 monomer (httex1) undergoes a conformational change to acquire a misfolded conformation (httex1*), which may then revert to the original conformation or react with another httex1 molecule to form a dimer. The formation of a httex1 dimer is assumed to be irreversible, rapidly leading to the formation of larger aggregates. (In Model 2, the dimerization step is between two misfolded monomers httex1*.)
Figure 2: FACS sorting and fluorescence microscopy of cell populations were used to determine rates of aggregate formation.
(a) At 24 h after transfection with httex1-GFP, transiently transfected ST14A cells were sorted into populations with narrow httex1-GFP concentration distributions by FACS. Sample data and collection gates are shown. Httex1-GFP concentrations were estimated using GFP standard beads and a mean cell diameter of 19 
m. (b) Rates of aggregate formation were measured by fluorescence microscopy of isolated cell populations over time. Images shown are cells expressing httex1Q72-GFP. Time shown is hours after sorting. The formation of intracellular aggregates is a stochastic process. White scale bar in the zero-hour image, 10 m.
In the development of mathematical models describing httex1-GFP nucleation kinetics (see Supplementary Methods online for derivation), we considered several alternative limiting cases. First, we consider the case in which neither misfolding nor dimerization is rate-limiting, and a folded httex1 monomer can form an irreversible aggregate nucleus with a misfolded monomer:
(Model 1)
where Pno agg is the probability that a given cell will not have formed an aggregate at time t, Vcell is the volume of the cell, NA is Avogadro's number, kmisfold is the rate constant for misfolding of the native form, k+ is the rate constant for addition of an httex1 molecule to a misfolded molecule to form an aggregate nucleus, krefold is the rate constant for conversion of misfolded httex1 to its normal conformation and Chttex1 is the intracellular concentration of httex1. We also consider the limit where formation of a dimer is the rate-limiting step, which results in
(Model 1a)
where Kn* is the equilibrium constant for interconversion of httex1 between normally folded and misfolded forms. We next consider the misfolding event as the rate-limiting step, resulting in
(Model 1b)
Finally, we consider a model where only the collision of two misfolded monomers results in commitment to visible aggregate formation:
(Model 2)
Note that Models 1a and 2 show identical second-order dependence on httex1 concentration; however, the mechanistic interpretation of the constant coefficient differs substantially in the two models.
We determined the kinetics of aggregate formation in transiently transfected ST14A cells15 (conditionally immortalized rat striatal neurons) to test for consistency with each of these relationships; fluorescence microscopy provides a means to track stochastic events in cultured cell populations16. We transfected cells with httex1Qn-GFP, with n varying among 25, 47 or 72 glutamines (Q25, Q47 and Q72, respectively). The expression of httex1-GFP after transient transfection increases gradually over the first 
18 h before leveling off. At 24 h after transfection, we used fluorescence-activated cell sorting (FACS) to separate cells based on GFP intensity, which is proportional to httex1Qn-GFP concentration, assuming cells are roughly the same volume. The percentage of cells with aggregates in each cell population was then determined by fluorescence microscopy, and determined again one or more times thereafter (Fig. 2).
The fraction of cells that contained aggregates immediately after sorting was excluded from further analysis, as these cells had formed aggregates during the period of variable expression of the httex1-GFP transgene, and the presence of aggregates may affect the fluorescence measured during FACS.
We investigated the time-dependence of the aggregation process over the first 18 h after FACS sorting (Fig. 3a). During this period, cellular httex1-GFP levels remained approximately constant (see Supplementary Fig. 1 online). The probability of a cell remaining aggregate-free was found to drop exponentially with time for both Q47 and Q72 constructs, at all expression levels examined. Exponential decay of the number of unaffected cells has been observed for a variety of neurodegenerative processes1 and is characteristic of a single-hit event with a constant hazard function17. Of direct relevance to the experiments reported here, a simple analytical theory has been derived for the distribution of lag times for a stochastic nucleated polymerization reaction18, which matches the experimentally observed lag-time histograms for sickle hemoglobin aggregation in vitro19. After an initial transient, this theory predicts an exponentially distributed probability of nucleation lag time (Supplementary Methods), consistent with the exponential distribution observed here (Fig. 3a).
Figure 3: Rate of httex1-GFP aggregate formation as a function of time, concentration and polyglutamine length.
(a) Natural log of the fraction of cells not containing aggregates as a function of time (time after FACS sorting is shown on graph), showing exponential decline. (b) Concentration-dependence of the rate constant 
for aggregate formation. The form of the nucleation rate constant for each of the models under consideration is as follows: =VcellNAk+Kn*C2httex1 (Model 1a); =VcellNAkmisfoldC2httex1 (Model 1b); =VcellNAk+Kn*2C2httex1 (Model 2). The solid line is the least-squares fit to the leading constant for a second-order concentration-dependence ( = 
Chttex12; Models 1a and 2). The dotted line is the least-squares fit to a first-order concentration-dependence ( = 
Chttex1; Model 1b). Error bars denote 95% confidence intervals, n = 95 cells on average for each point in a, n = 647 cells on average for each point in b.
Aggregate appearance rates for populations with different httex1-GFP concentrations were calculated from data taken over the first 18 h after sorting (Fig. 3b). Regression analysis indicates that the data best fit a second-order concentration-dependence for aggregate formation, with only a single parameter being fit (the lumped constant VcellNAk+Kn* for Model 1a; VcellNAk+Kn*2 for Model 2). The fit for second-order concentration dependence (Models 1a and 2) is significantly better than that for first-order concentration dependence (Model 1b; P < 0.05, F = 2.22, d.f. = 23), and the fit for Model 1 was not significantly better than that for second-order concentration dependence (F = 1.1, not significant at P = 0.35). Therefore, the probability of a cell forming an httex1 aggregate is best described by Models 1a and 2, implying that the formation of a dimer, rather than the misfolding rate, is the rate-limiting step in nucleation for the visible aggregate formation process. This result is notable given that httex1 concentration in this experimental system is probably considerably higher than neuronal httex1 concentrations in vivo; at lower values of Chttex1, the dimerization process should dominate the observed kinetics of aggregate formation even more strongly. The value of k+Kn* (Model 1a) or k+Kn*2 (Model 2) calculated from the fit in Figure 3b is 4.3 
10-8 M-1 s-1 for Q47 and 8.5 10-7 M-1 s-1 for Q72.
The second-order concentration dependence of aggregate appearance is consistent with rapid equilibrium between the misfolded conformation of htt and its properly folded conformation, with the formation of a dimer being the rate-limiting step in aggregation. This is in agreement with previous work reporting that a misfolded polyQ monomer is the critical nucleus3, 20. The parameters for monomer stability and dimer formation have recently been measured for Q47 peptides in vitro20: k+ = 11,400 M-1 s-1 and Kn* = 2.6 10-9, resulting in k+Kn* = 3 10-5 M-1 s-1 (in Model 1a), or k+Kn*2 = 7.7 10-14 M-1 s-1 (in Model 2). Our measured value for this coefficient in cells for httex1Q47-GFP is 4.3 10-8 M-1 s-1; which would be 1,000-fold smaller than the in vitro parameters for Model 1a, or 1,000,000-fold bigger for Model 2.
The particular mechanistic interpretation of the parameter k+Kn* (Model 1a) or k+Kn*2 (Model 2) is confounded by several aspects of the intracellular environment: chaperone binding, proteasomal degradation of growing aggregates or nuclei, cytoplasmic or nuclear compartmentalization, and macromolecular crowding. Crowding markedly accelerates the aggregation of synuclein21 and human apolipoprotein C-II22, but retards homogeneous nucleation lag times for sickle cell hemoglobin by three to four orders of magnitude23. Clearly macromolecular crowding can exert either substantial accelerating or decelerating effects on protein aggregation processes in vivo, but the detailed effects cannot be predicted at this point for huntingtin24. Proteasomal degradation of misfolded nuclei before progression to visible aggregates would have the effect of adding a degradation term to the denominator of equation S2 (Supplementary Methods), such that Kn* in Models 1a and 2 would become
This would have the effect of decreasing the apparent value of k+Kn* measured.
In cultured neurons the appearance of inclusion bodies decreases the probability of cell death somewhat5, and a transgenic mouse with visible aggregates but no neurodegeneration has been constructed25. Regardless of whether the aggregates are themselves toxic, their formation is strongly correlated with polyglutamine length and httex1 concentration in the same fashion as cell death in culture5, 26. Furthermore, treatments that diminish aggregation also reduce toxicity in cell culture27, 28, 29. Consequently, visible aggregate formation can serve as a useful proxy for the kinetic events that underlie the toxicity of expanded polyglutamine htt. The role of neuronal dysfunction versus cell death in Huntington disease is also the subject of debate, with some evidence that the underlying neurodegeneration results from cell-cell interactions30; so aggregate nucleation may result in dysfunction of the cell in which it forms, causing death of nearby cells, rather than death of the cell forming the aggregate.
The rate measured here for aggregate formation of httex1Q47-GFP is approximately consistent with the time scale observed for neurodegeneration in patients with Huntington disease, if the concentration of httex1 is 100 nM, or on the order of 200,000 molecules of httex1 per neuron (Fig. 4). We could not identify any reports in the literature of an experimental estimate for the number of mhttex1 molecules in a striatal neuron in vivo. In light of the importance of this parameter for both design of therapeutic interventions and fundamental understanding of the mechanisms of Huntington disease onset and progression, it is to be hoped that it will be determined soon as the cataloging of proteomic inventories progresses. The potential effect of therapeutics such as RNAi31 that downregulate htt expression, or of binding molecules that sequester mhttex1 into a form resistant to aggregation29, is also shown in Figure 4 as a series of curves of progressively lower mhttex1 concentration. Because the probability of aggregate formation is second order in mhttex1 concentration (Models 1a and 2), such treatments might be efficacious even if the mhttex1 concentrations were reduced by only a factor of two.
Figure 4: Simulated fraction of neurons that do not experience an aggregate nucleation event as a function of concentration (10, 30, 100 or 300 nM) and age for httex1Q47, based on Model 1a or 2, and assuming neurons of volume equivalent to cultured ST14A cells.
The dashed lines bracket the age range 31–35 years, the expected age of onset for Huntington disease in individuals with Q47 expansion38.The therapeutic implications of downregulating htt or otherwise sequestering mhttex1 into a conformation resistant to misfolding are demonstrated by simulating a range of concentrations. The probability of aggregate formation is clearly a strong function of mhttex1 concentration, and so therapeutic interventions that only modestly lower mhttex1 might substantially extend the time until a meaningful number of aggregation events occur.
Not all clinical data on Huntington disease are consistent with the concentration-dependence observed in our study. Data for individuals with polyglutamine lengths >70 are limited, but the mean age of onset tends to be around age 10 years for this group (about one-fourth the age for individuals with 47 glutamines)32. We found that the rate constant for aggregate formation for httex1Q72-GFP was 20 times that for httex1Q47, which should reflect an age of onset for individuals with 72 glutamines in httex1 that was one-twentieth that for individuals with 47 glutamines. The discrepancy may be due to the high uncertainty in the age of onset for the group with juvenile Huntington disease. It is also possible that differences in plasticity as the brain finishes development and ages could have a role in disease progression. Another inconsistency is the observation that individuals homozygous for the expanded repeat gene, and who might express up to twice as much of the toxic protein, show nearly the same age of onset for the disease as those who are heterozygous33, 34. If the concentration of pathogenic httex1 fragments is actually twice as high in homozygotes, our model would predict that the exponent would be four times greater for the exponential drop in aggregate-free cell fraction. However, regulation of expression and degradation of htt from multiple alleles has not been well studied, so the actual concentration of mhttex1 fragments in homozygotes as compared with heterozygotes is unknown, and the aggregation substrate may not be at double the concentration in homozygotes. There is also substantial variability in the age of onset in Huntington disease and this observation is based on very few Huntington disease homozygote cases. These individuals do have worse symptoms after onset, which implies some increase in neurodegeneration35.
Interaction of a misfolded mhttex1 molecule with other, non-htt proteins would also proceed in a stochastic fashion similar to that observed here; therefore cellular dysfunction or death resulting from such interactions would be expected to occur with first-order dependence on mhttex1 concentration. Our model cannot exclude the possibility of such 'toxic interactions' of the misfolded protein underlying disease progression; however, interactions of other proteins with mhtt in its normal conformation would result in constant dysfunction and would not be consistent with the rare degeneration observed. The toxic mhtt species may then be the misfolded protein itself, the aggregated nucleus, or an intermediate further on in the pathway leading to the formation of an aggregate. The data presented here do not shed light on these questions, as only visible aggregate formation was measured.
The agreement between Models 1a and 2 and the experimental data suggests that commitment to visible aggregation occurs in living cells as a second-order process, dependent on collision of two htt molecules. Stochastic mathematical modeling of aggregate nucleation seems quantitatively consistent with the delay time for disease onset in patients, consistent with the hypothesis that aggregate nucleation is the proximal causal event underlying the pathogenic process in Huntington disease. Any alternate hypotheses should also be consistent with the long-delayed onset and random degeneration of neurons, and in particular must be quantitatively consistent with the occurrence of very rare events that drive the process.
Methods
Plasmid construction, cell culture, transfection and sorting.
Httex1Qn-GFP plasmids were created by subcloning XbaI/XhoI fragments from p416/PQn36 into pcDNA3.1(–) (Invitrogen), which is expressed from a CMV promoter. The exon I fragments contained 25, 47 or 72 glutamines.
Undifferentiated ST14A cells36 were cultured according to published protocols. One day after plating, cells were transiently transfected with the appropriate httex1Qn-GFP plasmid using Lipofectamine 2000 (Invitrogen). Twenty-four hours after transfection, cells were sorted on a MoFlo flow cytometer (Cytomation) on the basis of their GFP fluorescence. Populations of cells expressing different levels of httex1Qn-GFP were sorted into separate wells of a 96-well plate (Nalge Nunc International). Three thousand cells from each population were collected, and the plate was then incubated for 30 min to allow adhesion to the plate.
Fluorescence microscopy.
After sorting, each population of cells collected was examined by fluorescence microscopy. Cells were visualized using a 20 objective, an Axiovert 100 fluorescence microscope (Carl Zeiss International) and DeltaVision softWoRx software (Applied Precision, LLC). The fraction of cells with aggregates was determined by counting the number of cells containing at least one aggregate and the number of cells without aggregates. When each population of cells had been counted, the 96-well plate was returned to the incubator and counted again one or more times. For time courses of httex1-GFP expression and aggregation, time-lapse microscopy was used on cells cultured in an environmental chamber.
As the cells were sorted 24 h after transfection, some cells contained aggregates immediately after sorting. Because the presence of aggregates may confound the measurement of httex1-GFP concentration, and because these aggregates formed during a period of variable httex1-GFP concentration during the first day after transfection, the fraction of cells with aggregates at this time was determined and excluded from further analysis.
Httex1-GFP concentrations were calculated using BD FACS EGFP standard beads (Becton Dickinson). The fluorescence of the standard beads, non-transfected cells and sorted cell populations was determined using a MoFlo FACS machine (Cytomation). Total httex1-GFP per cell in each sorted population was determined by subtracting background autofluorescence of non-transfected cells and comparing the value to those obtained for the standard beads. The diameter of trypsinized cells was measured on a Deltavision Nikon microscope with softWoRx image analysis software and was found to be 19 m on average. Mean cell volume was thus determined to be 3.6 10-12 liters, assuming spherical cells. Total httex1-GFP was divided by the cell volume to give the final httex1-GFP concentration. The concentration of httex1-GFP was calculated assuming that all cells had the same volume. This assumption may result in an error of up to two-fold variation in the concentration of cells isolated; however, the cells within different populations collected by FACS show the same mean size and amount of variation, so the relative mean concentration should not be biased by this assumption.
Curve fits and statistics were obtained using KaleidaGraph (Synergy Software). Mathematical models were evaluated using the F test and statistical analysis was performed as described37. Estimates of the aggregation rate constant were only obtained for samples in which more than 10% of the cells showed aggregates by the end of the 18-h sample time, to avoid large variability from counting very small numbers of cells. This criterion excluded submicromolar expression levels of httex1Q47-GFP or httex1Q72-GFP, and no rate constant was available at any expression level for httex1Q25-GFP.
Note: Supplementary information is available on the Nature Chemical Biology website.
