Asynchronous oscillations due to antigenic variation in Malaria Pf

Transcription

1 Asynchronous oscillations due to antigenic variation in Malaria Pf Jonathan L. Mitchell and Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX SIAM LS, Pittsburgh, 21

2 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

3 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

4 Delays in disease Physical origins + Latency time between compartments. Incubation time. Infectious time. Temporary Immunity. + Transit time of biological process. Modeling + Constant coefficient ODEs: exponential distribution of immune times. Easy to analyze. + Integro-differential Es: arbitrary distributions. Hard to analyze. + Delay DEs: step distributions. All immune for the same fixed time τ.

5 Delays in disease Physical origins + Latency time between compartments. Incubation time. Infectious time. Temporary Immunity. + Transit time of biological process. Modeling + Constant coefficient ODEs: exponential distribution of immune times. Easy to analyze. + Integro-differential Es: arbitrary distributions. Hard to analyze. + Delay DEs: step distributions. All immune for the same fixed time τ.

6 Delay induced oscillations ODE: x(t) = rx(t) + Let x(t) exp(λt). + Characteristic equation: λ = r. + There exists a single real value λ, implying exponential growth or decay. DDE: x(t) = rx(t τ) + Let x(t) exp(λt). + Characteristic equation: λ = re λτ. + Let λ = σ + iω σ = re στ cos(ωτ), ω = re στ sin(ωτ) + Transcendental equations with multiple solutions + Allows for oscillatory solutions to a first-order DDE.

7 Delay induced oscillations ODE: x(t) = rx(t) + Let x(t) exp(λt). + Characteristic equation: λ = r. + There exists a single real value λ, implying exponential growth or decay. DDE: x(t) = rx(t τ) + Let x(t) exp(λt). + Characteristic equation: λ = re λτ. + Let λ = σ + iω σ = re στ cos(ωτ), ω = re στ sin(ωτ) + Transcendental equations with multiple solutions + Allows for oscillatory solutions to a first-order DDE.

8 Malaria Map

9 Malaria Life Cycle Inter-host vs. Intra-host Blood cycle Parasitized RBCs rupture 1-3 new parasites. Parasite generations lead to fever, etc. PRBCs avoid splenic removal by cytoadhering to arterial walls. Must attack with immune response. Antibodies and T-Lymphocytes recognize antigens displayed on PRBCs.

10 Plasomodium Falciparum Four strains of malaria in humans. P. vivax is the most common. P. falciparum is the most dangerous. + Highest parasite load in host. + Cytoadhering leads to clogging of arteries in cerebrum. cerebral malaria + Leading cause of death in humans by malaria

11 Antigenic variation in Pf Evade the host s IR and prolonged infection by changing the dominate genetic variant. + Parasite varies the major epitope on antigen PfEMP1. + Epitope: binding sites for immune response effectors. In the population there are 6 variants defined by unique major epitopes + An individual will have < 6 (1-2) variants. + Variants will share minor epitopes. Individuals exhibit switching (oscillations) of the dominant variant. + Sequential dominance. + Prevents IR from maintaining a prolong attack against a single variant. + Evolutionary survival strategy.

12 Antigenic variation in Pf Evade the host s IR and prolonged infection by changing the dominate genetic variant. + Parasite varies the major epitope on antigen PfEMP1. + Epitope: binding sites for immune response effectors. In the population there are 6 variants defined by unique major epitopes + An individual will have < 6 (1-2) variants. + Variants will share minor epitopes. Individuals exhibit switching (oscillations) of the dominant variant. + Sequential dominance. + Prevents IR from maintaining a prolong attack against a single variant. + Evolutionary survival strategy.

13 Antigenic variation in Pf Evade the host s IR and prolonged infection by changing the dominate genetic variant. + Parasite varies the major epitope on antigen PfEMP1. + Epitope: binding sites for immune response effectors. In the population there are 6 variants defined by unique major epitopes + An individual will have < 6 (1-2) variants. + Variants will share minor epitopes. Individuals exhibit switching (oscillations) of the dominant variant. + Sequential dominance. + Prevents IR from maintaining a prolong attack against a single variant. + Evolutionary survival strategy.

14 Antigenic variation in Plasomodium Falciparum Molecular switching mechanisms in a single cell are known. Coordination of the parasite population is not well understood. Recker et al. proposed an interaction between the variants via the minor epitopes. + Switching occurs as a natural dynamic of the hosts IR. + No external switching mechanism or rule is needed. Recker et al., Nature (24) 429:

15 Antigenic variation in Plasomodium Falciparum Molecular switching mechanisms in a single cell are known. Coordination of the parasite population is not well understood. Recker et al. proposed an interaction between the variants via the minor epitopes. + Switching occurs as a natural dynamic of the hosts IR. + No external switching mechanism or rule is needed. Recker et al., Nature (24) 429:

16 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

17 Model of Recker and Gupta Bull. Math. Bio (26) 68: Y j : variant j parasitized red-blood cells. Z j : variant j specific immune response. W j : cross-reactive immune response affecting variant j.

18 Model of Recker and Gupta Parasitized RBCs: proliferation - removal due to IR. dy j dt = φy j αy j Z j α Y j W j Variant specific IR: stimulation - natural degradation. dz j dt = βy j T µz j Cross-reactive IR: multi-variant stimulation - natural degradation. dw j dt = β k ξ jk Y k T µ W j Delayed activation of IR (Mitchell & Carr) Y k T = Y k (t T )

19 Some assumptions Specific IR (z) is long lived relative to the cross-reactive IR (w). < µ µ 1 Complete sharing of minor epitopes global coupling. ξ jk Y k T with ξ jk = 1 k n Y k T k=1

20 Some assumptions Specific IR (z) is long lived relative to the cross-reactive IR (w). < µ µ 1 Complete sharing of minor epitopes global coupling. ξ jk Y k T with ξ jk = 1 k n Y k T k=1

21 Steady states Disease free: (Y j, Z j, W j ) = (,, ). Unstable. Nonuniform: (Y j, Z j, W j ). Unstable. Uniform: (Y j, Z j, W j ) = (Y, Z, W ). Stable.

22 Rescale and nondimensionalize New variables are deviations from the uniform steady-state (y j, z j, w j ) = (,, ) a = dy j dt dz j dt dw j dt = (z j + w j )(1 + y j ) = c n y j τ az j = 1 n n y k τ abw j, k=1 dµ φ, b = µ µ, c = αβ µφ α β and τ = d T. < µ µ 1

23 Synchronous vs. Asynchronous Synchronous: y j (t) = y(t), etc. 1 n n y k τ = y(t) k=1 Asynchronous: y j (t) y k (t), etc The plan... + Synchronous linear stability + Asynchronous linear stability + Asynchronous nonlinear dynamics

24 Synchronous vs. Asynchronous Synchronous: y j (t) = y(t), etc. 1 n n y k τ = y(t) k=1 Asynchronous: y j (t) y k (t), etc The plan... + Synchronous linear stability + Asynchronous linear stability + Asynchronous nonlinear dynamics

25 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

26 Synchronous linear stability Decay: oscillatory or monotonic? γ = α /α µ =.4 µ =.1 γ l γ u (b) y y 1.5 γ = 1.5 (a) γ = (b) (a) γ = (c) t µ = variant specific IR death rate << 1 y.2 (c)

27 Decay: oscillatory or monotonic? γ α α = removal rate due to cross-reactive IR removal rate due to specific IR If γ is sufficiently large or small then there are oscillations. Decreasing (increasing) the number of shared of minor epitopes n, shifts both critical values up (down). µ can be set such that there are always decaying oscillations. The variant-specific IR can be quite slow, while still being large enough to guarantee oscillations.

28 Decay: rates [( )( )] EZ + E W µ 1/2 Decay rate ab =, φ E W E Z αβ µ and E W α (nβ ) µ. E Z,W = efficacy of the specific and cross-reactive IR. The farther away one moves from the triangular region the variants oscillate with faster decay. Increasing the specific efficacy relative to the cross-reactive efficacy leads to faster decay.

29 Decay: rates [( )( )] EZ + E W µ 1/2 Decay rate ab =, φ E W E Z αβ µ and E W α (nβ ) µ. E Z,W = efficacy of the specific and cross-reactive IR. The farther away one moves from the triangular region the variants oscillate with faster decay. Increasing the specific efficacy relative to the cross-reactive efficacy leads to faster decay.

30 Delayed IR λ 3 + a(1 + b)λ 2 + a 2 bλ + e λτ [(1 + q)λ + a(1 + qb)] =. T h = 1 ( ) Ez + E w. φ Parasite generation rate φ T h. System is more susceptible to delay induced oscillations. E w E z E W T h. Decreases the sensitivity of the system. E z E W T h 1/φ. Thus, just as a strong parasite generation rate and a strong cross-reactive IR lead to decaying oscillations in the case of instantaneous IR, they also decrease the minimum value of delay necessary to excite persistent oscillations.

31 Delayed IR λ 3 + a(1 + b)λ 2 + a 2 bλ + e λτ [(1 + q)λ + a(1 + qb)] =. T h = 1 ( ) Ez + E w. φ Parasite generation rate φ T h. System is more susceptible to delay induced oscillations. E w E z E W T h. Decreases the sensitivity of the system. E z E W T h 1/φ. Thus, just as a strong parasite generation rate and a strong cross-reactive IR lead to decaying oscillations in the case of instantaneous IR, they also decrease the minimum value of delay necessary to excite persistent oscillations.

32 Delayed IR λ 3 + a(1 + b)λ 2 + a 2 bλ + e λτ [(1 + q)λ + a(1 + qb)] =. T h = 1 ( ) Ez + E w. φ Parasite generation rate φ T h. System is more susceptible to delay induced oscillations. E w E z E W T h. Decreases the sensitivity of the system. E z E W T h 1/φ. Thus, just as a strong parasite generation rate and a strong cross-reactive IR lead to decaying oscillations in the case of instantaneous IR, they also decrease the minimum value of delay necessary to excite persistent oscillations.

33 Delayed IR λ 3 + a(1 + b)λ 2 + a 2 bλ + e λτ [(1 + q)λ + a(1 + qb)] =. T h = 1 ( ) Ez + E w. φ Parasite generation rate φ T h. System is more susceptible to delay induced oscillations. E w E z E W T h. Decreases the sensitivity of the system. E z E W T h 1/φ. Thus, just as a strong parasite generation rate and a strong cross-reactive IR lead to decaying oscillations in the case of instantaneous IR, they also decrease the minimum value of delay necessary to excite persistent oscillations.

34 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

35 Asynchronous linear stability 3 n system of equations. dy j dt dz j dt dw j dt = (z j + w j )(1 + y j ) = c n y j τ az j = 1 n n y k τ abw j, k=1 Characteristic equation with 3 n roots. [F 1 (λ)f ap (λ, τ)] n 1 F s (λ, τ) = F 1 (λ) = λ + ab F ap(λ, τ) = λ 2 + aλ + c n e λτ F s(λ, τ) = λ 3 + a(1 + b)λ 2 + a 2 bλ + e»λ λτ 1 + c «+ a 1 + bc «. n n

36 Asynchronous linear stability 3 n system of equations. dy j dt dz j dt dw j dt = (z j + w j )(1 + y j ) = c n y j τ az j = 1 n n y k τ abw j, k=1 Characteristic equation with 3 n roots. [F 1 (λ)f ap (λ, τ)] n 1 F s (λ, τ) = F 1 (λ) = λ + ab F ap(λ, τ) = λ 2 + aλ + c n e λτ F s(λ, τ) = λ 3 + a(1 + b)λ 2 + a 2 bλ + e»λ λτ 1 + c «+ a 1 + bc «. n n

37 Sync vs. Antiphase eigenvectors [F 1 (λ)f ap (λ, τ)] n 1 F s (λ, τ) = n 1 roots from F 1. Always stable. 3 roots from F s. + Same as synchronous case with synchronized eigenvector v j = v. 2(n 1) roots from F ap. + ap = antiphased eigenvectors n j=1 v (y) j = v (y) jm = ei2πjm/n,

38 Sync vs. Antiphase eigenvectors [F 1 (λ)f ap (λ, τ)] n 1 F s (λ, τ) = n 1 roots from F 1. Always stable. 3 roots from F s. + Same as synchronous case with synchronized eigenvector v j = v. 2(n 1) roots from F ap. + ap = antiphased eigenvectors n j=1 v (y) j = v (y) jm = ei2πjm/n,

39 Sync vs. Antiphase eigenvectors [F 1 (λ)f ap (λ, τ)] n 1 F s (λ, τ) = n 1 roots from F 1. Always stable. 3 roots from F s. + Same as synchronous case with synchronized eigenvector v j = v. 2(n 1) roots from F ap. + ap = antiphased eigenvectors n j=1 v (y) j = v (y) jm = ei2πjm/n,

40 Decay rates, NO DELAY.2 (a) Synchronous.2 (b) Antiphase y j z j w j Time, t Time, t Antiphase: Decay rates: synchronous vs. asynchronous σ s 1 2 µ faster than σ ap 1 2 µ

41 Long-time observation is async: NO DELAY Variants, y j Variants, y j Variants, y j.1.1 (a) Time,t.1 (b) Time,t.2 (c) Time,t I(u 1 ) I(u 2 ) I(u 3 ) R(u 1 ) R(u 2 ) R(u 3 ) Given an arbitrary initial condition... Complex oscillations can be decomposed into a sum of synchronous and antiphase oscillatory modes... The synchronous component decays fast... Observe some combination of antiphase oscillations... observe asynchronous oscillations.

42 Long-time observation is async: NO DELAY Variants, y j Variants, y j Variants, y j.1.1 (a) Time,t.1 (b) Time,t.2 (c) Time,t I(u 1 ) I(u 2 ) I(u 3 ) R(u 1 ) R(u 2 ) R(u 3 ) Given an arbitrary initial condition... Complex oscillations can be decomposed into a sum of synchronous and antiphase oscillatory modes... The synchronous component decays fast... Observe some combination of antiphase oscillations... observe asynchronous oscillations.

43 Long-time observation is async: NO DELAY Variants, y j Variants, y j Variants, y j.1.1 (a) Time,t.1 (b) Time,t.2 (c) Time,t I(u 1 ) I(u 2 ) I(u 3 ) R(u 1 ) R(u 2 ) R(u 3 ) Given an arbitrary initial condition... Complex oscillations can be decomposed into a sum of synchronous and antiphase oscillatory modes... The synchronous component decays fast... Observe some combination of antiphase oscillations... observe asynchronous oscillations.

44 Linear stability: τ Hopf bifurcation to persistent oscillations. Synchronous: Antiphase T s T ap = 1 φ = 1 φ ( Ez + E w E w ( Ez + E w E z ). ),

45 Sync vs. Antiphase: τ µ Antiphase (a) (b) Synchronous µ y j y j.5 (a) (b) Time, t Increasing µ weakens specific IR + Cross-reactive IR specific IR Couples variants synchronous. Increasing µ weakens cross-reactive IR + Specific IR cross-reactive Decouples variants asynchronous. slope = nα β αβ

46 Sync vs. Antiphase: τ µ Antiphase (a) (b) Synchronous µ y j y j.5 (a) (b) Time, t Increasing µ weakens specific IR + Cross-reactive IR specific IR Couples variants synchronous. Increasing µ weakens cross-reactive IR + Specific IR cross-reactive Decouples variants asynchronous. slope = nα β αβ

47 Hopf bifurcation to asynchronous oscillations Near Hopf point. τ = τ h + ǫ 2 τ 2. Multiple time scales t and s = ǫ 2 t. Expand y = ǫy (1) + ǫ 2 y (2) +... Expand the delay term: y j (t τ, s ǫ 2 τ) = y j τh ǫ 2 ( τ 2 y j t Validity of expansion not guaranteed. y j + τ h τh s τh ) +O(ǫ 4 ),

48 Hopf bifurcation to asynchronous oscillations Near Hopf point. τ = τ h + ǫ 2 τ 2. Multiple time scales t and s = ǫ 2 t. Expand y = ǫy (1) + ǫ 2 y (2) +... Expand the delay term: y j (t τ, s ǫ 2 τ) = y j τh ǫ 2 ( τ 2 y j t Validity of expansion not guaranteed. y j + τ h τh s τh ) +O(ǫ 4 ),

49 Hopf bifurcation to asynchronous oscillations Near Hopf point. τ = τ h + ǫ 2 τ 2. Multiple time scales t and s = ǫ 2 t. Expand y = ǫy (1) + ǫ 2 y (2) +... Expand the delay term: y j (t τ, s ǫ 2 τ) = y j τh ǫ 2 ( τ 2 y j t Validity of expansion not guaranteed. y j + τ h τh s τh ) +O(ǫ 4 ),

50 Hopf bifurcation to asynchronous oscillations Near Hopf point. τ = τ h + ǫ 2 τ 2. Multiple time scales t and s = ǫ 2 t. Expand y = ǫy (1) + ǫ 2 y (2) +... Expand the delay term: y j (t τ, s ǫ 2 τ) = y j τh ǫ 2 ( τ 2 y j t Validity of expansion not guaranteed. y j + τ h τh s τh ) +O(ǫ 4 ),

51 Antiphase oscillations as basis The leading order, O(ǫ) problem is linear. Y (1) = J τh Y (1), t Solution decomposed as a sum of the antiphase eigenvectors. x (1) j y (1) j w (1) j = iω h y (1) + e.d.t., j = n 1 X A m(s)v jm e iω h t + c.c. + e.d.t., m=1 = + e.d.t., A m (s), m = 1, 2,...,n 1 are slowly varying amplitudes. Determined by solvability condition at O(ǫ 3 ). da m ds = τ 2(f 2 + ig 2 )A m + (f 3 + ig 3 )Âm + (f 4 + ig 4 )ÃnA n m,

52 Two examples for n = 3 (a) Pure antiphase with A 1, A 2 = (b) Combination of basis A 1 = A z k y k w k.1 (a) x Scaled Time.1 (b) x Scaled Time y max (a) τ ap τ b τ y max.5.4 (b) τ ap τ b τ

53 Two examples for n = 3 (a) y 2 s f 2 (τ τ h ) f 3 cos θ(t) + 2π 3 cos θ(t) + 4π 3 cos (θ(t) + ) 1 C A (b) y = 2 s f 2 (τ τ h ) f 3 + 2f A cos θ(t). s y max φez 6 (T Tap), E z + E w µ z k y k w k.1 (a) x Scaled Time.1 (b) x Scaled Time y max (a) τ ap τ b τ y max.5.4 (b) τ ap τ b τ φ or E Z larger amplitude.

54 Transient and persistent chaotic oscillations y j τ = Time, t τ = Time, t τ = Time, t

55 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

56 Summary: synchronous oscillations Key model assumptions: + Variant specific + cross-reactive IR sequential dominance. + Variant specific µ cross-reactive µ. Synchronous oscillations: + Identify IR efficacies as useful parameters. E Z αβ µ and E W α (nβ ) µ. + A large parasite generation rate and a strong cross-reactive IR favors oscillations. + Increases the sensitivity to persistent oscillations due to external forces such as a delayed IR. Pulsating solutions Y for long times. Poorly timed measurements of the system could be misleading.

57 Summary: sync. vs async. oscillations Asynchronous oscillations = antiphase. Synchronous: decay rate E W and is fast. Antiphase: decay rate E Z and is slow. Given arbitrary ICs, the likely observation is asynchronous oscillations. The frequency of async. is higher than synch. Forces the immune system to respond faster. Inc/dec E W relative to E Z strengthens/weakens coupling. + Strong coupling: synchronous oscillations. + Balanced coupling: sequential dominance. + Very weak coupling: uncoordinated oscillations.

58 Open questions Less than complete set of minor variants. Dynamics on network. Stronger physiologically based model.

59 Outline Introduction Modeling Synchronous oscillations Asynchronous oscillations Summary Additional material

60 Model of Recker and Gupta Recker et al., Nature (24) 429: Recker and Gupta, Bull. Math. Bio (26) 68: De Leenheer and Pilyugin, J. Biological Dynamics (28) 2:12-12 Mitchell and Carr, Bull. Math. Bio. (29) 72:59-61 Blyuss and Gupta, J. Math. Biol. (29) 58: Mitchell and Carr, submitted

61 Warning! Taylor series with delay can be misleading From R.D. Driver, Ordinary and Delay Differential Equations x = 2x(t) + x(t τ) Let x = e λt λ = 2 + e λτ σ + 2 = e στ cos(ωτ), ω = e στ sin(ωτ) Consider the real-part equation σ + 2 e στ σ< 2 1 σ+2 στ e σ < : Exponentially decaying solutions 1 στ e

62 Small delay: τ 1 x = 2x(t) + x(t τ) x = 2x(t) + [x(t) τx (t) τ 2 x (t) +...] Let x = e λt and keep O(τ 2 ) λ = 2 + [1 τλ τ 2 τ 2 ] 1 2 τ 2 τ 2 (τ + 1)λ + 1 = λ = (τ + 1) ± (τ + 1) 2 2τ 2 τ 2 λ + > for all τ: Exponentially growing solutions. Must validate analytical results with numerical simulations.