1/30 Stochastic Models for Yeast Prion Propagation Diana Cole 1, Lee Byrne 2, Byron Morgan 1, Martin Ridout 1, Mick Tuite Institute of Mathematics, Statistics and Actuarial Science, University of Kent 2. Department of Biosciences, University of Kent, Funded by BBSRC

2/30 Contents 1.Introduction to Prions 2.The Process of Curing 3.Cell Reproduction 4.Improvements to Curing Model 5.Is Cell Reproduction Required for Curing? 6.Conclusions

3/30 1. Introduction to Prions Prion = Infectious Protein Particle Mammal prion is the infectious agent in neuro- degenerative diseases eg BSE, CJD Protein exists in two forms: PrP C – normal PrP SC – abnormal, infectious form Abnormal form can cause conversion of normal form protein to abnormal form Yeast prions have many similar properties, and are much easier to study

4/30 Yeast prion is an abnormal form of the protein SUP35p Gives rise to 2 types of yeast cell (phenotypes) [psi-] – all SUP35p is in the normal form [PSI+] – some or all SUP35p in the abnormal form (Prion form) Abnormal protein forms prion seeds or propagons Distinguish by a colour assay [psi-] [PSI+]

5/30 3mM GdnHCl added to flask at t = 0 Samples taken every hour Plates left for 2-3 days. Count no. white and red colonies 2. The Process of Curing

6/30 [PSI+] [psi  ]

7/30 If Guanidine hydrochloride (GdnHCl) is added to [PSI+] cells, the cells gradually revert to [psi-] (Eaglestone et al, 2000). GdnHCl blocks prion replication. At cell division the prions are distributed between the resulting cells. The number of prions in each cell is diluted over time, so that the population gradually reverts to [psi-].

8/30 g = 0 g = 1 g = 2 g = 3

9/30 Simple Dilution Model (Eagelstone et al, 2000) N – number of prions present in a cell N ~ Bin(N(0),2  g ) (where g is the generation number) Curing probability p + (g|N(0)) = 1  {1  2  g } N(0) Assume that N(0) ~ Po(n 0 )  p + (g) = 1  exp(  n 0 2  g ) Need to estimate generation number g = t /  (where  is mean cell reproduction time). Note this assumes that all cells take  hours exactly to divide.

10/30 Fitting simple dilution model n 0 = (16.6)

11/30 Model Improvements Incorporating variation in the time that cells take to divide Yeast cells don’t divide equally Unequal prion distribution between mother and daughter cells

12/30 3. Cell Reproduction Population size grows exponentially E{T(t)}  exp(  t) (where  is Malthusian parameter or growth rate)  = 1.18  = 0.41

13/30 Malthusian Parameter (  ) (equal cell division i.e E. coli) From theory of age dependent branching processes 1 = 2 E{exp(-M  )} (M time a cell takes to divide) By assuming there is no variation get approx  = ln(2) /  (  mean time a cell takes to divide) Better approximations from Cowan (1985) and Ridout et al (2006) Ridout et al (2006) approx where CV is coefficient of variation (=  /  ) Doubling time, t d, is the time the population takes to double in size. t d = ln(2) /  Simple approx (no variation) t d =  Ridout et al approx

14/30 E. coli Time-Lapse

15/30 Mean = CV = M ~ Gamma( ,  )  =  /   = 0.44 (0.02)  = 33.5 (6.9)  tdtd simple Ridout et al exact

16/30 Malthusian Parameter (  ) (unequal cell division i.e s. cerevisiae) S. cerevisiae yeast cell divide by budding Mother cells take M hours to divide Daughter cells take an extra D hours to divide (M +D hours in total) 1 = E{exp(-M  )}[1+ E{exp(-D  )}] (Green, 1981) Ridout et al (2006) approx where Daughter Mother

17/30 S. Cerevisiae Time-Lapse

18/30 [PSI+] and [psi-] cells take about the same time to divide Cells reproduce slower in the presence of GdnHCl and ratio D/M is much larger MothersDaughters  MeanSDnmeanSDnSimpleNewExact [PSI+] -GdnHCl [psi-] -GdnHCl [PSI+] +GdnHCL [psi-] +GdnHCl

19/30 Generation Number As the cells in a population all have been dividing at different times, it is not possible to determine the generation number of an individual cell. Instead we examine the expected generation number, G. Current approx Cole et al (2007) approx Asymmetric cell division

20/30 4. Improvements to the Curing Model Variation p + (t) = 1  exp(  n 0 2  g ) where Morgan et al (2003) develop a stochastic version of this model. where Q g denotes the expected number of cells at generation. g is found by assuming that cell M ~ Gamma( ,  ) with mean  =  / 

21/30 Asymmetric Cell Division and Unequal Prion Segregation Cole et al (2004) model  is probability prion is passed to daughter cell n 0 is average number of prions at start Q g,d (t) is the expected number of cells at time t that have had d daughter cell divisions out of g generations. Calculated by assuming that M ~ Gamma(  M,  ) with mean  M =  M /  and D ~ Gamma(  D,  ) with mean  D =  D / .

22/30 Over-parameterised Model Need to estimate 5 parameters: n 0, ,  M,  D, . Can only estimate 2 parameters. (Cole et al, 2004 shows model is nearly parameter-redundant). Time-lapse gives mother and daughter cells division times. Curing data also gives total no. of cells, T(t). Slope of ln{T(t)} gives us one cell division parameter. (Ridout et al, 2006). Adhoc approach. Use q =  D /  M and  from time-lapse. Obtain  M from T(t). Estimate n 0 and  from curing data. Combined likelihood l = l C + l TL + l T(t). However assumes independence, but l C and l T(t) are not. Simulation shows combined likelihood approach is biased at estimating n 0 and , whereas adhoc approach isn’t.

23/30 Fitting Cole et al (2004) Model EstSE n 

24/30 5. Is Cell Reproduction Required for Curing? Wu et al (2005) claim that curing of [PSI+] by GdnHCl does not require cell division

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26/30 Cell death Cells die with probability p If p is constant and using model of Green (1981)  and G are determined by the mean, the variability and the probability of cell death. If certain conditions reduce the growth rate, then this could mean the cells are reproducing slower or that the conditions are causing cell death (or a mixture of both). Typically cells die when they are old, and as a population of cells is largely made up of young cells the effect is minimal Alpha factor causes more cell death.

27/30 Alpha-factor Experiments

28/30 EstimateSE n 0 (combined) lag (  -factor)  (  -factor)  (no  -factor)

29/30 6. Conclusion Cell division is critical to the process of curing (Byrne et al, 2007) Important to include variability, and asymmetric cell division in curing model. Daughter cells get less prions than mother cells at cell division. Variability, asymmetric cell division and cell death have an effect on cell reproduction. Current approximations for Malthusian parameter and generation number are not always accurate enough.

30/30 References Byrne, L. J., Cox, B. S., Cole, D. J., Morgan, B. J. T, Ridout, M. S. and Tuite, M. F. (2007) The role of cell division in curing the [PSI+] prion of yeast by guanidine hydrochloride Proceedings of the National Academy of Sciences of the United States of America, 104, Cole, D.J., B.J.T. Morgan, M. S. Ridout, L.J. Byrne and M.F. Tuite. (2004) Estimating the number of prions in yeast cells. Mathematical Medicine and Biology 21, Cole. D.J., B.J.T. Morgan, M.S. Ridout, L.J. Byrne and M.F. Tuite. (2007) Approximation for Expected Generation Number Biometrics doi: /j x Cowan, R. (1985). Branching Process Results in Terms of Moment of Generation-Time Distribution. Biometrics 41, Eaglestone, S.S., Ruddock, L.W., Cox, B.S. and Tuite, M.F. (2000). Guanidine hydrochloride blocks a critical step in the propagation of the prion-like determinant [PSI+] of saccharomyces cerevisiae. Proceedings of the National Academy of Sciences of the United States of America, 97, Green (1981). Modelling Yeast Cell Growth Using Stochastic Branching Processes. Journal of Applied Probability, 18, Morgan, B.J.T., Ridout, M.S. and Ruddock, L. (2003) Models for yeast prions. Biometrics, 59, M.S. Ridout, D.J. Cole, B.J.T. Morgan, L.J. Byrne and M.F. Tuite. (2006) New approximations to the Malthusian parameter Biometrics, 62, Wu Y., Greene L. E., Masison, D. C. and Eisenberg E. (2005) Curing of yeast [PSI+] prion by guanidine inactivation of Hsp1-4 does not require cell division. Proceedings of the National Academy of Sciences of the United States of America, 102,