Modeling prion dynamics (In progress)

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1 Modeling prion dynamics (In progress) Peter Olofsson and Suzanne Sindi Trinity University and Brown University Mathematics Department Division of Applied Mathematics August 21, 2012

2 PRIONS Infectious agents composed of misfolded protein (no DNA or RNA).

3 PRIONS Infectious agents composed of misfolded protein (no DNA or RNA). Mad cow disease, Creutzfeldt Jakob disease.

4 PRIONS Infectious agents composed of misfolded protein (no DNA or RNA). Mad cow disease, Creutzfeldt Jakob disease. Studied in yeast (does no harm).

5 !"#$%&'(' 1,.#'0(''(*# Transmission Synthesis )*#+&,'(*# Conversion Fragmentation -,./0&#$.$(*# Figure: Yeast Prion Cycle. There are four steps essential for the persistence of the prion state: synthesis, conversion, fragmentation and transmission from mother to daughter cell.

6 !"#$%&'(' 1,.#'0(''(*# Transmission Synthesis )*#+&,'(*# Conversion Fragmentation -,./0&#$.$(*# Figure: Yeast Prion Cycle. There are four steps essential for the persistence of the prion state: synthesis, conversion, fragmentation and transmission from mother to daughter cell. (Note to Kimmel: Branching within branching!)

7 CURING CURVE: The fraction of cells that have prions as a function of time. Starts at 100%, declines toward 0%. Many prions initially means slower decline. One goal: estimate number of prions in initial cell from curing curve.

8 SOME PREVIOUS WORK: L.J. Byrne, D.J. Cole, B.S. Cox, M.S. Ridout, B.J.T. Morgan, and M.F. Tuite. The number and transmission of [PSI+] prion seeds (propagons) in the yeast Saccharomyces cerevisiae. PLoS One, 4(3):4670, DJ Cole, BJT Morgan, MS Ridout, LJ Byrne, and MF Tuite. Estimating the number of prions in yeast cells. Mathematical Medicine and Biology, 21(4):369, 2004.

9 SOME PREVIOUS WORK: L.J. Byrne, D.J. Cole, B.S. Cox, M.S. Ridout, B.J.T. Morgan, and M.F. Tuite. The number and transmission of [PSI+] prion seeds (propagons) in the yeast Saccharomyces cerevisiae. PLoS One, 4(3):4670, DJ Cole, BJT Morgan, MS Ridout, LJ Byrne, and MF Tuite. Estimating the number of prions in yeast cells. Mathematical Medicine and Biology, 21(4):369, No conversion (prions do not grow). Tends to underestimate initial number of prions since larger prions are more difficult to pass on to daughter.

10 DISCRETE MODEL: Binary splitting, no death. After splitting: one mother, one daughter.

11 DISCRETE MODEL: Binary splitting, no death. After splitting: one mother, one daughter. Z n = number of cells with prions in nth generation.

12 DISCRETE MODEL: Binary splitting, no death. After splitting: one mother, one daughter. Z n = number of cells with prions in nth generation. Fraction of cells with prions: P n = E[Z n] 2 n

13 DISCRETE MODEL: Binary splitting, no death. After splitting: one mother, one daughter. Z n = number of cells with prions in nth generation. Fraction of cells with prions: P n = E[Z n] 2 n A given prion is transmitted to the daughter cell with probability p < 0.5.

14 Cell in nth generation has ancestry of the type of d daughters and m mothers. MMDDM...MMDM

15 Cell in nth generation has ancestry of the type of d daughters and m mothers. MMDDM...MMDM Without prion growth: Initial prion present with probability p d (1 p) m.

16 Cell in nth generation has ancestry of the type of d daughters and m mothers. MMDDM...MMDM Without prion growth: Initial prion present with probability p d (1 p) m. With prion growth: easier to be in MDMMM than in MMMMD.

17 Cell in nth generation has ancestry of the type of d daughters and m mothers. MMDDM...MMDM Without prion growth: Initial prion present with probability p d (1 p) m. With prion growth: easier to be in MDMMM than in MMMMD. Prions grow one unit at a time according to a Poisson process with rate β.

18 Assume a critical size after which prions can no longer be transmitted to the daughter cell.

19 Assume a critical size after which prions can no longer be transmitted to the daughter cell. Initial cell is i units (conversion events) from critical size.

20 Assume a critical size after which prions can no longer be transmitted to the daughter cell. Initial cell is i units (conversion events) from critical size. Consider sequences in generation n where the final daughter is in position k and there is a total of l daughters. There are ( k 1 l 1 ) such sequences.

21 Assume a critical size after which prions can no longer be transmitted to the daughter cell. Initial cell is i units (conversion events) from critical size. Consider sequences in generation n where the final daughter is in position k and there is a total of l daughters. There are ( k 1 l 1 ) such sequences. Example: n = 5, k = 3, l = 2 : DMDMM, MDDMM

22 Assume a critical size after which prions can no longer be transmitted to the daughter cell. Initial cell is i units (conversion events) from critical size. Consider sequences in generation n where the final daughter is in position k and there is a total of l daughters. There are ( k 1 l 1 ) such sequences. Example: n = 5, k = 3, l = 2 : DMDMM, MDDMM Probability p nkl such a sequence has prions? Depends on critical generation G when initial prion gets too big. From G on, prion stays in mother.

23 Assume a critical size after which prions can no longer be transmitted to the daughter cell. Initial cell is i units (conversion events) from critical size. Consider sequences in generation n where the final daughter is in position k and there is a total of l daughters. There are ( k 1 l 1 ) such sequences. Example: n = 5, k = 3, l = 2 : DMDMM, MDDMM Probability p nkl such a sequence has prions? Depends on critical generation G when initial prion gets too big. From G on, prion stays in mother. If G < k, p nkl = 0 and if G = j k, p nkl = p l (1 p) j l.

24 Hence where p nkl = n 1 p l (1 p) j l P i (G = j) + p l (1 p) n l P i (G n) j=k P i (G = j) = H i (j + 1) H i (j) where H i is the distribution function for the gamma distribution with parameters i and β.

25 Now assume N i initial prions of size i. Conditional probability at least one is in given sequence: so that P N i nkl = 1 (1 p nkl) N i where ϕ is the pgf of N i. P nkl = 1 ϕ(1 p nkl )

26 All taken together: expected fraction of cells with prions is P n = 2 n [ n k=0 k l=0 ( ) ] k 1 (1 ϕ(1 p l 1 nkl ))

27 All taken together: expected fraction of cells with prions is P n = 2 n [ n k=0 k l=0 ( ) ] k 1 (1 ϕ(1 p l 1 nkl )) Obvious extension to many different i and N i : multivariate pgf s.

28 100 simulations, 5th and 95th percentiles (black), estimated curve (red), large β (fast growth) and small β (slow growth). Probability of Prion Aggregates Probability of Prion Aggregates

29 Initial number of prions n 0 = 209, our estimate n 0 = 191.8, disregarding prion growth n 0 = Probability of Prion Aggregates

30 CONTINUOUS MODEL: CMJ, not BH

31 CONTINUOUS MODEL: CMJ, not BH Newborn cell needs time T to grow and mature, produces first daughter cell at time D, then produces daughter cells at times D + M 1, D + M 1 + M 2,... Reproduction process ξ(dt) = δ D (dt) + N k=1 δ D+M1 + M k (dt) where N = total number of daughter cells (can practically assume N = ) [see Green (1981)].

32 CONTINUOUS MODEL: CMJ, not BH Newborn cell needs time T to grow and mature, produces first daughter cell at time D, then produces daughter cells at times D + M 1, D + M 1 + M 2,... Reproduction process ξ(dt) = δ D (dt) + N k=1 δ D+M1 + M k (dt) where N = total number of daughter cells (can practically assume N = ) [see Green (1981)]. Malthusian parameter determined by 0 ( e αt F D (dt) + e αt F M (dt) ) = 1 Expressions similar to but much more complicated than discrete case.

33 CONTINUOUS MODEL: CMJ, not BH Newborn cell needs time T to grow and mature, produces first daughter cell at time D, then produces daughter cells at times D + M 1, D + M 1 + M 2,... Reproduction process ξ(dt) = δ D (dt) + N k=1 δ D+M1 + M k (dt) where N = total number of daughter cells (can practically assume N = ) [see Green (1981)]. Malthusian parameter determined by 0 ( e αt F D (dt) + e αt F M (dt) ) = 1 Expressions similar to but much more complicated than discrete case.

34 where P(t) E[Z t] E[Y t ] E[Y t ] = 1 F A (t) + + n=1 n k=0 ( n 1 k 1 [( ) n 1 ( F A F n 1,k 1(t) F A F n,k(t)) k ) ( F A F n 1,k(t) F A F n,k(t)) ] and E[Z t ] = n n=0 k=0 k l=0 ( ) k 1 [1 ϕ(1 p l 1 nkl (t)))]

35 One detail: how does it start?

36 One detail: how does it start? Cole et al: Initial cell has excess life distribution, sampled from a renewal process. Pdf: f A (t) = q D 1 µ D (1 F D (t)) + q M 1 µ M (1 F M (t))

37 One detail: how does it start? Cole et al: Initial cell has excess life distribution, sampled from a renewal process. Pdf: f A (t) = q D 1 µ D (1 F D (t)) + q M 1 µ M (1 F M (t)) We: initial cell has stable age distribution, sampled from exponentially growing population. Pdf: f A (t) = α e αs (f D (s + t) + f M (s + t))ds 0 Note presence of Malthusian parameter α. Any practical relevance? Don t know.

38 Grant support: NIH grant 1 F32 GM (Sindi) NIH grant 1 R15 GM (O)

Branching process models of prion dynamics

Branching process models of prion dynamics Peter Olofsson, Suzanne Sindi, Jason Davis Trinity University and UC Merced Mathematics Applied Mathematics April 7, 2015 PRIONS Infectious agents composed of