Supplementary Information
The Physical Basis of How Prion Conformations
Determine Strain Phenotypes
Motomasa Tanaka1,2, Sean R. Collins1, Brandon H. Toyama1 & Jonathan S. Weissman1
1Howard Hughes Medical Institute, Department of Cellular and Molecular Pharmacology, University of California-San Francisco and California Institute for Quantitative Biomedical Research, San Francisco, California 94143, USA
2PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi,
Saitama 332-0012, Japan
1. Methods
(1) Yeast strains and reagents
Throughout in vivo experiments, we used isogenic [psi-][RNQ+] and [PSI+] derivatives of 74D-694 [MATa, his3, leu2, trp1, ura3; suppressible marker ade1-14(UGA)]1. [PSI+(Sc4], [PSI+(Sc37] and [PSI+(SCS)] strains were made by infection of [psi-] with in vitro produced Sup-NM Sc4, Sc37 and SCS amyloid fibres2. Sc4, Sc37 and SCS (previously termed as Sc[Ca3[Sc4]]) fibres were produced as described previously2. Sup-NM proteins carboxy-terminally tagged with 7x-histidine were expressed in E. Coli. (DE3) and purified as reported previously3.
(2) In vitro analysis of strain conformations of Sup-NM amyloids
Sup-NM amyloid fibres were prepared by dilution of the denatured proteins into buffer A (5mM potassium phosphate buffer, 150mM NaCl, pH7.4) in the absence or presence of 5% (mol/mol) seed. Sup-NM SCS amyloid fibres were formed at 23°C with 5% (mol/mol) seed of SCS amyloids. The AFM-based single fibre assay was performed as reported previously4, with the exception that Sup-NM fibre seeds produced by spontaneous polymerization were directly used without passage through multiple rounds of polymerization and reseeding. Each histogram involved the measurement of at least 150 individual fibres. The fibre growth rate of distinct Sup-NM amyloid fibres was also examined by thioflavin T fluorescence, as reported previously2, in the absence or presence of Ficoll PM70 (Amersham Biosciences) to the final concentration of 25% (wt/v). For fibre rigidity assay, Sup-NM amyloid fibres were formed by polymerizing Sup-NM (5M) in a beaker (2.2cm diameter), containing 2ml of buffer A, overnight under undisturbed conditions in the presence of 5% (mol/mol) seed of Sc4, Sc37 and SCS amyloids at 4, 37 and 23 °C, respectively. The amyloid solution was stirred by a magnetic stir bar (3mm diameter, 1cm length) at ~100 rpm and an aliquot was taken at 0, 30, 60 min and used for morphological, seeding efficacy and prion infectivity analyses. Images of the Sup-NM fibres before and after the stirring were acquired by AFM (Digital Instruments, MultiMode AFM, Nanoscope software). Seeding efficacy of the Sup-NM amyloids before and after the stirring was examined by thioflavin T fluorescence, as reported previously2.
(3) In vivo analysis of prion strains
Protein infection experiments were performed as reported previously3. The curing assay of [PSI+] with guanidine hydrochloride (Gdn) was performed following the procedure by Cox, Tuite and coworkers5,6. Logarithmic cultures of [PSI+(Sc4)], [PSI+(Sc37)] and [PSI+(SCS)] strains were grown in YPAD+3mM Gdn for 12-15 hours, transferred to YPAD media and cultured for 0, 0.75, 1.5, 2.5 and 3.5 hours. The cultures were then returned to YPAD+3mM Gdn media and samples were taken at 2~4-hour intervals over ~30-40 hours. Percentage of [PSI+] cells were determined by plating ~200 cells on 1/4 YEPD plates. The yeast generations were calculated using a hemocytometer and colony counts on 1/4 YEPD plates at each sample time. The percentages of [PSI+] cells are normalized with that at the time when they were put into or returned to YPAD+3mM Gdn media and to 100% [PSI+] cells at this time of rescue; the fraction of [psi-] colonies that were in the culture at the time of rescue was subtracted from the percentage of [psi-] colonies at each time point. Average numbers of prions per cell were determined by fitting the plot of percentage of [PSI+] cells against cell generation with the equation, y=100*(1-exp(-A*(2^(-x)))), where x, y, and A indicates cell generation, percentage of [PSI+] cells and an average number of prions, using IgorPro5.0 (WaveMetrics Inc.). The procedure of counting propagons was performed as described previously5. An aliquot of YEPD cultures of [PSI+(Sc4)], [PSI+(Sc37)] and [PSI+(SCS)] strains was placed on YEPD+3mM Gdn plates and single cells were transferred to different locations on the plate with a dissection microscope. Following ~30 hr of growth, a piece of agar containing whole individual colonies was resuspended with water and plated on SD-Ade or SD trace Ade (0.4 mg/L) plates. 1/4YEPD and YEPD+3mM Gdn plates were further used to distinguish [PSI+] colonies from Ade+ revertants. The number of [PSI+] colonies was counted for more than 40 single cells for each [PSI+] strain. The “jackpot” cells containing >200 propagons were excluded in calculation of the mean value. For agarose gel analysis, yeast cells were lysed in buffer (25mM Tris (pH7.5), 50mM KCl, 10mM MgCl2, 1mM EDTA, 5% glycerol, 1mM PMSF, protease inhibitor cocktail (Roche)) with glass beads and the crude lysate was partially clarified by centrifugation at 10,000g for 5 minutes. The lysates (200g) were treated with SDS sample buffer (50mM Tris (pH6.8), 5% glycerol, 2% SDS, 0.05% bromphenol blue) at 37°C for 10 minutes. The size of prion particles was analyzed by horizontal 1.6% agarose gels in TAE buffer containing 0.1% SDS7, followed by immunoblotting with a polyclonal anti Sup-NM antibody1. Chicken pectoralis extracts including titin (3000 kDa), nebulin (750 kDa) and myosin heavy chain (200 kDa) as well as BenchMark Pre-Stained Protein Ladder (Invitrogen) were used to estimate molecular weight. Sedimentation analysis was performed as described previously3, except for the speed and time of ultracentrifugation being 30,000 rpm and 10 minutes, respectively.
2. Analysis of the model describing prion strains
(1) Analysis of steady-state solutions for a single prion strain:
Our model, as described in the text and Fig. 1a, yields the following differential equations describing the time-evolution of the concentration of soluble Sup35p ([x]), prion fibres ([y]), and aggregated Sup35p ([z]), where rate constants for translation of Sup35p, growth of prion fibres, division of prion fibres per unit length, and cellular growth are given by α, , γ, and R, respectively:
d[x]/dt = α – [x][y] – R[x]
d[y]/dt = γ[z] – R[y]
d[z]/dt = [x][y] – R[z]
We wish to find steady-state solutions to these equations to understand the possible stable (or meta-stable) states of the cell. This system of equations is composed of two separable systems, one of which describes the total concentration of Sup35p ([xtot] = [x] + [z1] + [z2]):
d[xtot]/dt = α – R[xtot]
This requires that [xtot] = α/R at steady-state and leaves a system of two equations for our steady-state solutions:
d[y]/dt = γ[z] – R[y] = 0
d[z]/dt = (α/R – [z])[y] – R[z] = 0
This gives two steady-state solutions. One occurs when [y] = [z] = 0 and [x] = α/R. The other occurs at
[x] = R2/γ
[y] = αγ/R2 – R/
[z] = α/R – R2/γ
We note that this second steady-state solution is physically meaningful only if all concentrations are greater than or equal to zero. From both the equations for [y] and [z], we then see that this second solution is physically relevant and distinct from the first solution only for αγ > R3.
We are also interested in whether these steady-state solutions are stable or unstable. We can determine this by computing the eigenvalues (λ) of the Jacobian at each steady-state point. The Jacobian (in [y] and [z]) can be written as:
The characteristic equation for the eigenvalues (λ) is then given by
λ2 + λ([y] + 2R) + (–γ[x] + R[y] + R2) = 0
For the first steady-state, this equation becomes
λ2 + 2Rλ + (–γ[x] + R2) = 0
Solving the quadratic equation then yields
λ = –R ± sqrt(αγ/R)
When a [PSI+] solution exists (αγ > R3), this gives one positive and one negative eigenvalue, so the [psi-] solution will be unstable. However, when there is no [PSI+] solution (αγ < R3), this solution will be stable.
For the second ([PSI+]) solution, we can see that both eigenvalues are real because the discriminant
([y] + 2R) 2 – 4(–γ[x] + R[y] + R2)
which can be rewritten as
2[y]2 + 4γ[x]
is greater than zero (since [x] > 0 and [y] > 0). Additionally, both eigenvalues are negative because their sum (given by –1[y1] – 2R) is less than zero and their product is positive. The product (given by –γ[x] + R[y] + R2) is positive because, substituting in the steady-state values for [x] and [y], it can be rewritten as
αγ/R – R2
which is greater than zero because αγ > R3 for any physically relevant [PSI+] solution. Thus the [PSI+] solution is stable and the [psi-] solution is unstable.
(2) Potential elaborations of the model:
Several additional features could have been included in our model that we did not focus on in the present analysis because either evidence suggests they are not significant effects, or, as described below, they would likely only represent minor perturbations to the amount of soluble Sup35p or the number of fibres per cell and would not change the general conclusions of the present paper. Nonetheless, as discussed below, some of these elaborations can and will alter other observable features of prion strains (most notably the rate of spontaneous loss of the prion state as a function of average fibre number per cell) and thus the below considerations should help provide a framework for future efforts to make quantitative models that predict such features:
Fibre clumping: Comparison between the number of independently inheritded propogan with the apparent fibre length measured using the agarose gel assay suggests that individual Sup35p fibres may laterally associate in vivo7. In some cases, very large clumps of fibres have been found to be non-transmissible dead-end aggregates, although these are rare in rapidly dividing cells in the absence of overexpression8. As long as the extent of association is not so large as to block the accessibility of the fibres to monomers and chaperones, the underlying processes of growth and division should be relatively unaffected by lateral association of fibre. Only the mitotic stability will be reduced because the number of propagons per cell will be the number of fibres per cell divided by the number of fibres per fibre clump. This point is illustrated by recent studies of prion propagation in yeast containing an Hsp70 chaperone mutation, SSA1-21 which was reported to increase the clumping of Sup35p polymers by an order of magnitude9. While the mutation has minor effects on the fibre length and amount of soluble Sup35 as measured by the agarose gel assay, it induces observable sectoring of [PSI+] colonies due to a decrease in propagon number. This result indicates that even under conditions in which mutations in the chaperone machinery artificially cause a large increase in fibre clumping, the process of fibre growth and division, which determine the length of the fibres and the amount of soluble Sup35p, are not significantly affected. Thus, to the extent that lateral association is likely to occur in the absence of over-expression of Sup35p or defects in the chaperone machinery, fibre clumping should not strongly impact the fibre growth and division process which is being modeled and monitored in the present work.
End-to-end annealing of fibres: The work of DePace, et al.4 demonstrated that Sup35p fibres do not anneal in vitro to any detectable degree, and there is no evidence to suggest that they do in vivo.
Dissolution of small fibres and/or release of monomers after fibre division or fibre depolymerization: Evidence from in vitro4 and in vivo6,10 experiments suggests that such dissolution does not occur to a significant degree. Additionally, in a stochastic version of our model which we explored through computer simulation (see below), we included the dissolution of very small fibres formed after fragmentation and found little deviation from the predictions derived above.
Stochastic effects from cell to cell variability: We reasoned that stochastic effects resulting from the enclosure of prion propagation inside small cellular volumes could significantly affect some properties of the prion state. Therefore, we developed a discrete stochastic simulation model based on our continuous model presented above and also including the additional property that fibres formed after fibre division could dissolve if they were shorter than a minimum threshold length. As discussed below, the basic kinetics of growth and division, the average number of propagons per cell, and the average concentration of soluble Sup35p are not significantly affected. However, we do find that the rate of spontaneous loss of the prion state is significantly greater (often by several orders of magnitude) than would be expected from the mean number of propagons in a cell. This provides an explanation for the observation that finite observable rates of loss of [PSI+] are observed even when there are tens of propagons per cell. The difference arises because loss of the prion state generally occurs from a minority of cells which, due to stochastic events in cell division, are carrying a significantly smaller than average number of propagons. Therefore, the rate of loss of the prion state is expected to depend not only on the average number of propagons per cell, but also on the ability of a strain to recover rapidly from a temporary decrease in its number of propagons. This second property will be determined by sqrt(αγ/R) – R (as discussed below under “Recovery of a prion strain after inhibition of fibre division”).
We constructed a discrete stochastic model based on the continuous model presented above. The stochastic model was formulated as an executable Java program. In the stochastic model, the propagation of the prion was simulated in individual cells, keeping track of the number (and integer lengths) of all prion fibres in a cell, as well as the concentration of soluble Sup35p molecules. Cells were modeled to grow exponentially in size, and divide once they reached twice their original size. Fibres were allowed to break between any two adjacent monomers and fibres smaller than a minimum threshold size were assumed to spontaneously dissociate into monomers. Fibre growth and division were modeled as purely stochastic processes. For the purposes of simulation, we estimated α = 0.0154 μM∙min-1, R = 0.0077 min-1, a minimum fibre length of five monomers, and cellular volume varying between 1.66 and 3.32 x 10-14 L. We then varied and γ and compared our results to those predicted by the continuous model.
We found that the two models gave very similar patterns of estimates for the concentration of soluble Sup35p and the average number of fibres per cell. However, estimates of the rate of loss of the prion state that account for the variation seen in the stochastic simulations often differed dramatically from estimates computed from the continuous model that did not account for cell to cell variation (in which the rate is estimated to be 2μ-1, where μ is the mean number of fibres per cell). Therefore, an attempt to model accurately the rate of loss of the prion state must account for cell-to-cell variability and the relative ability of a strain to recover from temporary depletion of its pool of prion seeds.
Shown above are plots of the concentration of soluble Sup35p, the average number of fibres per cell, and a calculated rate of loss of the prion state where each point represents the results obtained with the continuous versus the stochastic model for one choice of parameter values. The blue lines in each plot indicate the trends that would have been observed if there was perfect correspondence between the two models.
(3) Competition between two strains:
Let [x] be the concentration of soluble Sup35p, [y1] and [z1] be the concentrations of fibres and aggregated Sup35p for strain 1, and [y2] and [z2] be the concentrations of fibres and aggregated Sup35p for strain 2. Let α be the rate of synthesis of new Sup35p, R be the rate of cellular growth, 1 and 2 be the rate constants for growth of fibres of strain 1 and strain 2, and γ1 and γ2 be the rate constants for fibre division for the two strains. Then we can extend our model to get the following equations:
d[x]/dt = α – 1[x][y1] – 2[x][y2] – R[x](1)
d[y1]/dt = γ1[z1] – R[y1](2)
d[z1]/dt = 1[x][y1] – R[z1](3)
d[y2]/dt = γ2[z2] – R[y2](4)
d[z1]/dt = 2[x][y2] – R[z2](5)
Steady-state will be achieved at values of [x], [y1], [z1], [y2], and [z2] such that all of the above derivatives are equal to zero. We can readily see that steady-state can be achieved if either [y1] = [z1] = 0 or [y2] = [z2] = 0 or both. The case where all four values are zero corresponds to the [psi-] case, which is identical to the single strain case. The case where [y2] = [z2] = 0, but [y1], [z1] > 0 corresponds to the prion state for strain 1 as would be found in the single strain analysis, and similarly [y1] = [z1] = 0 and [y2], [z2] > 0 corresponds to the prion state for strain 2. Furthermore, from equations (2 - 5) we see that a steady-state with [y1], [z1] > 0 and [y2], [z2] > 0 requires that
[x] = R[z1]/1[y1] = R[z2]/1[y2] and [z1]/[y1] = R/γ1 and [z2]/[y2] = R/γ2
Thus such a steady-state would require that
[x] = R2/1γ1 = R2/2γ2 or 1γ1 = 2γ2
Provided that 1γ1 ≠ 2γ2, this leaves only three steady-state solutions. As in the single strain case, the [psi-] solution is unstable. Now, the strain that will win the competition is the strain for which the corresponding steady-state solution is stable, and the strain that will lose the competition is the strain for which the corresponding steady-state solution is not stable. To find the stable steady-state, we evaluate the eigenvalues of the Jacobian at the steady-state points. At the steady-state corresponding to the prion state of strain 1,
[x] = R2/1γ1
[y1] = αγ1/R2 – R/1
[z1] = α/R – R2/1γ1
[y2] = [z2] = 0
As for the single strain case, the system of equations is composed of two separable systems, one of which describes the total concentration of Sup35p ([xtot] = [x] + [z1] + [z2]):
d[xtot]/dt = α – R[xtot]
We find that the total concentration of Sup35p at steady-state is equal to α/R and eliminate equation (1) to yield a system of four linearly independent equations describing the derivatives of the concentrations of [y1], [z1], [y2], and [z2], where
[x] = α/R – [z1] – [z2] = α/R – [z1](since [z2] = 0)
Taking advantage of the fact that [y2] and [z2] are zero, we find that the Jacobian of this system is given by:
where [y1] and [x] are given by the values above. The characteristic polynomial will then factor to give two equations from which the eigenvalues (λ) can be found. The first (from the upper lefthand block) is
λ2 + λ(1[y1] + 2R) + (–1γ1[x] + R1[y1] + R2) = 0
This equation is identical to the one obtained for the eigenvalues in the case of a single prion strain in the absence of competition, and so for values of 1 and γ1 that correspond to the existence of a prion state, it will yield two negative eigenvalues.
The other two eigenvalues are given by the other factor of the characteristic equation: