Statistical mechanics of lymphocyte networks modelled with slow and ...

Statistical mechanics of lymphocyte networks modelled with slow and fast variables Alexander Mozeika and Anthony CC Coolen Institute for Mathematical and Molecular Biomedicine, King’s College London.

June 17, 2016

Alexander Mozeika and Anthony CC Coolen

Statistical mechanics of lymphocyte networks

Outline

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



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.



Immune Response: Interaction of B cells and T cells



Immune Response: B cells, T helper cells and Ag

Results of flow cytometry on day 7 after immunization (Baumjohann et. al. 2013 Immunity 38 596).



Immune Response: Helper and Regulator T cells

Regulatory and helper T cells in the germinal centre response (Vanderleyden et. al. 2014 Arthritis Res. Ther. 16 471).



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. 2010 Blood 116 1070) and Single-cell RNA sequencing (Stubbington et. al. 2016 Nature Methods 13 329). On the other hand, mathematical models of clone-size distributions mainly use stochastic processes (Desponds et. al. 2016 PNAS 113 274) and ordinary differential equations (De Boer et. al. 2001 J. Theor. Biol. 212 333), 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.



Dynamics of B-clones Dynamics: B-clones, specified by the log-concentration b = (b1 , . . . , bM ), are governed by the Langevin equation τb

dbµ dt

where hχµ (t)χν (t 0 )i =

= Fµ (σ) − ρbµ + χµ (t) 2τb δµν δ(t−t 0 ) . β˜

P µ The signal Fµ (σ) = Jµ ξ σ + θ is a function of µ i i∈∂µ i T-clones specified by the concentrations σ = (σ1 , . . . , σN ). P The interaction Jµ = M ν=1 Sµν aµ , where a = (a1 , . . . , aM ) are “epitope” concentrations of Ag/Ags. The i-th T-clone is helper (regulator) if ξiµ > 0 (ξiµ < 0). T-independent activation of B-clones is facilitated by θµ . Alexander Mozeika and Anthony CC Coolen


(1)

Lymphocyte Network

Interactions of helper and regulator T-clones with B-clones. Alexander Mozeika and Anthony CC Coolen


Dynamics of T-clones The energy function H(b, σ) = − allows us to write dbµ τb dt

M X

µ=1

= −

M

1 X 2 bµ Fµ (σ) + ρ bµ 2

(2)

µ=1

∂ H(b, σ) + χµ (t). ∂bµ

(3)

We assume that the same energy function governs T-clones X ∂ dσi = Jµ ξiµ bµ − V (σ) + ηi (t), (4) τσ dt ∂σi µ∈∂i

where hηi (t)ηj (t 0 )i =

2τσ δij δ(t−t 0 ) . β



Fast equilibration of B-clones Assume that B-clones are fast variables (τb → 0) and 1 −βH(b, ˜ σ). P(b|σ) = e Z (σ)

Furthermore,

dσi dt

=−

D

E

∂ ∂σi H(b, σ)

= −

(5)

+ ηi (t)

∂ F(σ) + ηi (t), ∂σi

(6)

where F(σ) = −β˜−1 log Z (σ), from which follows P(σ) = Alexander Mozeika and Anthony CC Coolen

1 −βF (σ ) e . Z Statistical mechanics of lymphocyte networks

(7)

Fast equilibration of B-clones Joint distribution Pβ,β˜(b, σ)   F (σ ) 2  β PM  − 12 ρβ˜ bµ − µ ρ M F 2 (σ ) Y  e e 2ρ µ=1 µ q , (8) = β PM R ˜) Fµ2 (σ   µ=1 2ρ µ=1  ˜ Dσ ˜e 2π/ρβ where Dσ = e−βV (σ ) dσ. Distribution of B clone concentrations P(c) =

Z

2 − 12 ρβ˜ log(c)− Fρ

e


c

q 2π/ρβ˜

P(F ) dF .


(9)

Fast equilibration of B-clones

B clones create interactions, with strength Jijµ = helper and regulator T clones. Alexander Mozeika and Anthony CC Coolen

Jµ2 µ µ ρ ξi ξj ,

between


Analysis of equilibrium: Fast B-clone equilibration regime Assume that T clone: σi ∈ {−1, 1} (active regulator or helper), σi ∈ {0, 1} (active or inactive helper), etc. Then T clones are governed by the distribution PM

P(σ1 , . . . , σN ) =

e

µ=1

2 βJµ ρ

(

2 P i∈∂µ ξi σi 2

2 ( βJµ P PM µ=1 ρ e ˜ σ

)

2 P ˜i i∈∂µ ξi σ 2

)

.

(10)

Above is equivalent to ferromagnetic Ising model when σi ∈ {−1, 1} or σi ∈ {0, 1} with ξi = 1. P Average T-clone “activity”: m = N1 N i=1 hσi i gives us the fraction of helper, m+ = 1+m , and regulator, m− = 1 − m+ , 2 T clones. There are many T clone networks with finite βc (N → ∞) such that: m = 0 (m+ = 21 ) when β < βc and m 6= 0 (either m+ > 21 or m+ < 12 ) when β > βc . Alexander Mozeika and Anthony CC Coolen


Analysis of equilibrium

Examples of lymphocyte (B-clone and T-clone ◦) network topologies leading to T clone networks (right) with finite βc .




1−m 2 , and βJ 2 of ρµ .

Fraction of regulator T-clones, m− = T-clones , m+ =

1+m 2 ,

as a function


fraction of helper



˜

Average B clone size, hci = e1/2ρβ heF /ρ iβ , as a function of Alexander Mozeika and Anthony CC Coolen

βJµ2 ρ .





Systems on random regular graphs Assume: Each T-clone is “connected” to L B-clones and each L B clone is connected to K T-clones ( M N = K ). Assume: σi ∈ {−1, 1}, Jµ = J and θµ = 0. Recursive equation: hP PK −1 iL−1 2 1 βJ 2 PK −1 σj +σ ) +φ j=1 σj 2 ρ ( j=1 e {σj } P[σ] = P −1 iL−1(11) 2 1 βJ 2 PK −1 P hP σ ˜j +˜ σ ) +φ K ˜j j=1 σ 2 ρ ( j=1 e σ ˜ {˜ σj } 1 log(P[+1]/P[−1]) 2 Above canbe used to compute m± = φ =

m = tanh

Lφ L−1

P(F ) =

, and P

{σj } e

P

P 1 β 2 F +φ K j=1 2 ρ

1 βJ 2 PK ( j=1 ρ

2 {˜ σj } e


σj

1 2

(1 ± m), where

δ F −J σ ˜j )2 +φ

PK

j=1 σj

PK

j=1

σ ˜j

. (12)


Phase diagram



B-clone size distribution for L = K = 4 Statistical mechanics of lymphocyte networks 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

The distribution ofconnectivity B clone sizes studied three L = K = 4, was studied for for the B-clone noisedifferent parameters ˜ 2B-clone {0.5, 1.0, 2.0}, represented by the dotted, dashed and solid lines respectively, in the high < and low > right) ( ⇡ 0.0929) noise regimes. Top: noise parameters in the “low” (top andT-clone “high” (bottom left Left: The average B-clone size, hci, as a function of the fraction of T regulator cells , m . Right: The distribution P (c) of the B-clone size c for = 0.0639 m− = 0.1 and bottom right m = 0.9) “amount” of Ag regimes. − size distributions for = 0.1219 with m = 0.1 (m = ). Bottom: B-clone c

1 2

(left) and m

= 0.9 (right).


c

c


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:1603.01328 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. Alexander Mozeika and Anthony CC Coolen
