Statistical mechanics of lymphocyte networks modelled with slow and fast variables

Transcription

1 modelled with slow and fast variables Institute for Mathematical and Molecular Biomedicine, King s College London. June 17, 2016

2 Outline Adaptive Immune System Motivation Dynamics of clonal expansion Statistical mechanics of clonal expansion Results Summary

3 Adaptive Immune System Immune system (IS) defends organism from invading pathogens such as viruses, bacterium, parasites, etc. In complicated organisms usually divided into two parts: innate and adaptive IS. Innate IS is a first line of defence but nonspecific. Adaptive IS is more specific and offers a more long-term immunity by learning and memorising a wide range of pathogens.

4 Immune Response: Interaction of B cells and T cells

5 Immune Response: B cells, T helper cells and Ag Results of flow cytometry on day 7 after immunization (Baumjohann et. al Immunity ).

6 Immune Response: Helper and Regulator T cells Regulatory and helper T cells in the germinal centre response (Vanderleyden et. al Arthritis Res. Ther ).

7 Motivation One one hand, distributions of B cell and T cell clone-sizes can be obtained by modern experimental techniques such as High-throughput (Yu-Chang Wu et. al Blood ) and Single-cell RNA sequencing (Stubbington et. al Nature Methods ). On the other hand, mathematical models of clone-size distributions mainly use stochastic processes (Desponds et. al PNAS ) and ordinary differential equations (De Boer et. al J. Theor. Biol ), but usually do not consider interactions between B cells and T cells. In this work we first define model of interacting B clones and T clones then we use statistical mechanics to obtain distributions B-clone sizes.

8 Dynamics of B-clones Dynamics: B-clones, specified by the log-concentration b = (b 1,..., b M ), are governed by the Langevin equation τ b db µ dt = F µ (σ) ρb µ + χ µ (t) (1) where χ µ (t)χ ν (t ) = 2τ bδ µνδ(t t ). β ( ) The signal F µ (σ) = J µ i µ ξµ i σ i + θ µ is a function of T-clones specified by the concentrations σ = (σ 1,..., σ N ). The interaction J µ = M ν=1 S µνa µ, where a = (a 1,..., a M ) are epitope concentrations of Ag/Ags. The i-th T-clone is helper (regulator) if ξ µ i > 0 (ξ µ i < 0). T-independent activation of B-clones is facilitated by θ µ.

9 Lymphocyte Network Interactions of helper and regulator T-clones with B-clones.

10 Dynamics of T-clones The energy function allows us to write H(b, σ) = τ b db µ dt M b µ F µ (σ) + 1 M 2 ρ bµ 2 (2) µ=1 µ=1 = b µ H(b, σ) + χ µ (t). (3) We assume that the same energy function governs T-clones τ σ dσ i dt = µ i J µ ξ µ i b µ where η i (t)η j (t ) = 2τσδ ij δ(t t ) β. σ i V (σ) + η i (t), (4)

11 Fast equilibration of B-clones Assume that B-clones are fast variables (τ b 0) and P(b σ) = 1 Z(σ) e βh(b,σ). (5) Furthermore, dσ i dt = σ i H(b, σ) + η i (t) = σ i F(σ) + η i (t), (6) where F(σ) = β 1 log Z(σ), from which follows P(σ) = 1 Z e βf(σ). (7)

12 Fast equilibration of B-clones Joint distribution P β, β(b, σ) M e 1 2 ρ β ( = µ=1 2π/ρ β b µ Fµ(σ) ρ ) 2 where Dσ = e βv (σ) dσ. Distribution of B clone concentrations P(c) = e 1 2 ρ β c 2π/ρ β e β M 2ρ µ=1 F µ(σ) 2 D σ e β 2ρ ( ) 2 log(c) F ρ M µ=1 F 2 µ( σ), (8) P(F ) df. (9)

13 Fast equilibration of B-clones B clones create interactions, with strength J µ ij = J2 µ ρ ξµ i ξ µ j, between helper and regulator T clones.

14 Analysis of equilibrium: Fast B-clone equilibration regime Assume that T clone: σ i { 1, 1} (active regulator or helper), σ i {0, 1} (active or inactive helper), etc. Then T clones are governed by the distribution P(σ 1,..., σ N ) = e M µ=1 βj 2 µ ρ ( i µ ξ i σ i) 2 σ e M βjµ 2 ( i µ ξ i σ i) 2 µ=1 ρ 2 Above is equivalent to ferromagnetic Ising model when σ i { 1, 1} or σ i {0, 1} with ξ i = (10) Average T-clone activity : m = 1 N N i=1 σ i gives us the fraction of helper, m + = 1+m 2, and regulator, m = 1 m +, T clones. There are many T clone networks with finite β c (N ) such that: m = 0 (m + = 1 2 ) when β < β c and m 0 (either m + > 1 2 or m + < 1 2 ) when β > β c.

15 Analysis of equilibrium Examples of lymphocyte (B-clone and T-clone ) network topologies leading to T clone networks (right) with finite β c.

16 Analysis of equilibrium Fraction of regulator T-clones, m = 1 m 2, and fraction of helper T-clones, m + = 1+m 2, as a function of βj2 µ ρ.

17 Analysis of equilibrium Average B clone size, c = e 1/2ρ β e F /ρ β, as a function of βj2 µ ρ.

18 Analysis of equilibrium

19 Systems on random regular graphs Assume: Each T-clone is connected to L B-clones and each B clone is connected to K T-clones ( M N = L K ). Assume: σ i { 1, 1}, J µ = J and θ µ = 0. Recursive equation: [ {σj } e 1 βj 2 2 ρ ( K 1 j=1 σ j +σ) 2 +φ ] K 1 j=1 σ L 1 j P[σ] = [ σ { σj } e 1 2 φ = 1 2 log(p[+1]/p[ 1]) ( Lφ L 1 βj 2 ρ ( K 1 j=1 σ j + σ) 2 +φ ] K 1 j=1 σ L 1 (11) j Above can be used ) to compute m ± = 1 2 (1 ± m), where m = tanh, and P(F ) = {σ j } e 1 β 2 ρ F 2 +φ K j=1 σ j δ ( F J K j=1 σ ) j { σ j } e 1 2 βj 2 ρ ( K j=1 σ j ) 2 +φ K j=1 σ j. (12)

20 Phase diagram

21 B-clone size distribution for L = K = 4 modelled with slow and fast variables 23 hci P (c) m c P (c) P (c) c c Figure 9. Behaviour of B clones in the immune system with fast B-clone equilibration. The system, defined on a random regular factor-graph with connectivity L = K = 4, was studied for the B-clone noise parameters 2 {0.5, 1.0, 2.0}, representedbythedotted,dashedandsolidlinesrespectively,in the high < c and low > c ( c ) T-clone noise regimes. Top: Left: The average B-clone size, hci, asafunctionofthefractionoftregulator cells, m. Right: The distribution P (c) oftheb-clonesizecfor = (m = 1 ). Bottom: B-clone size distributions for = with m =0.1 2 (left) and m =0.9 (right). The distribution of B clone sizes studied for three different B-clone noise parameters in the low (top right) and high (bottom left m = 0.1 and bottom right m = 0.9) amount of Ag regimes.

22 Summary We used statistical mechanics to study dynamics of B clones and T clones interacting on networks. We considered a simple scenario when T clones are modelled by binary variables. Many results of our analysis are independent of the network topologies and qualitatively consistent with experimental observations. Assumption of random network topology allows us to compute distributions of B-clone sizes. Preprint is available at arxiv: Acknowledgements Biology: Deborah Dunn-Walters, Victoria Martin, Joselli Silva O Hare. Comp. Biology and Physics: Franca Fraternali, Alessia Annibale, Adriano Barra, Elena Agliari and Silvia Bartolucci.