the immune system from a statistical mechanics...

Performance and breakdown of the immune system from a

statistical mechanics perspective

Elena Agliari Università di Parma

INFN - Gruppo Collegato di Parma

Bari, SM&FT, 21-23 September 2011venerdì 23 settembre 2011

Statistical Mechanics and Theoretical Immunology

Immune system devoted to protect host body against damaging substances

Made up of a huge number of different kinds of cells and messengers, which must be properly

orchestrated to ensure safe performance

Pattern recognition, memory storage, self/non-self discrimination are cooperative features

Statistical mechanics Systemic viewpoint to evidence emergent

properties and the key mechanisms underlying (mis)functioning of the system

venerdì 23 settembre 2011

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B cells make up a “ferromagnetic network”Experimental facts

B lymphocytes are dichotomic →σ = ± 1Each lymph. carries a specificity →ξ={1,0,...,1}

Clone (ξi) ↔ Specific antibody (ξi)Several (N) different clones → {ξ1, ξ2,..., ξN}

Model 0System of NxM spins

N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]

Antigenic stimulation ↔ External local field hi

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Burnet’s (Nobel Prize 1960) clonal expansion theory

H0 = - cAg Σi hi mi

cAg is antigenic concentration

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N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]Exp. findings & “Small” repertoire → Functional network


N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]Antigenic stimulation ↔ External local field hi

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H0 = - cAg Σi hi mi

cAg is antigenic concentration

H1 = - 1/N Σi,j Jij mi mj

Jij measures affinity between clones i and j

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Jerne’s (Nobel Prize 1984) idiotypic network theory

[Abs also function as internal images of Ags]


The two approaches are synergic

H = H0 + H1 = - cAg Σi hi mi - 1/N Σi,j Jij mi mjArbitrary couple of nodes, i, j, the µ-th entries of corresponding stringsare complementary iff !µ

i != !µj " # complementary entries c # [0, L]

cij =!L

µ=1[!µi (1$ !µ

j ) + !µj (1$ !µ

i )] =!L

µ=1[!µi + !µ

j $ 2!µi !µ

j ]

Attractive vs Repulsive interaction! weight ! " 0f!,L("i, "j) # !cij $ (L$ cij) % [$L, !L]Nodes can interact pairwise according to a couplingJij ! !(f!,L(!i, !j)) exp[f!,L(!i, !j)]/J" Jij = exp[cij(" + 1)# L], whenever cij > L/(" + 1), otherwise Jij = 01/J normalizing pre-factor ensuring Jij has finite, unitary average $", L

Attractive vs Repulsive interaction → weight α ≥ 0

# Complementary matches

Minimal assumption & Uncorrelation among nodes! uniform distribution for !µ

i ’s extraction P (c) =!

12

"L !Lc

"

Probability that Jij > 0, i.e. i " j, is p!,L =#L

c=!L/(!+1)"+1 P (c)

Coupling distribution P (Jij) =$

(1/2)L! L(Jij+L)/(1+!)

"if J > 0

1# p if Jij = 0

Coordination number: zi =!N

j=1 Aij =!N

j=1!Jij"

Weighted coordination number: wi =!N

j=1 Jij

Coordination number: zi =!N

j=1 Aij =!N

j=1!Jij"

Weighted coordination number: wi =!N

j=1 Jij

Coupling distribution

Coordination number

Weighted Coordination number

➪ Ferromagnetic Correlated-Random-Bond Net

Idiotypic network [Varela et al.]

Cumulative frequency distributions of significant interactions (normalized optical densities) measured between a collection of 435 Abs [Carneiro et al. (1996)]


Cooperative features of the idiotypic network

- Low-Dose Tolerance- Bell-shape response- Memory Effects

Surviving self-reacting clones are quiescent due to the interaction with the network (over which information is spread)

Negative selection in bone marrow: most self-reacting clones are deleted

High weighted connectivity (low reactivity) group ➙ self addressed

Low weighted connectivity (high reactivity) group ➙ non-self addressed

Self-NonSelf Discrimination

Interaction matrix [Kearney et al.]venerdì 23 settembre 2011


- Low-Dose Tolerance- Bell-shape response- Memory Effects

Surviving self-reacting clones are quiescent due to the interaction with the network (over which information is spread)

Negative selection in bone marrow: most self-reacting clones are deletedSelf-NonSelf Discrimination

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

a

corre

latio

n co

effic

ient

Self=50Self=100Self=500Self=1000Self=5000

Simulation of negative selection yields to correlation between w and self-affinity

Negative selection is automatically accounted in our networkvenerdì 23 settembre 2011

Helper cells make the system complex

B lymphocytes. Play as APC

. Make specific Abs. Need signal of chemical

messengers from Th

Tk lymphocytes. Induce death of dangerous cells. Need signal of chemical messengers from Th

Th lymphocytes. No effector functions. Receive first signal from APC. Release/Absorb regulatory agents (cytokines)

Cytokines. Auto-, Para-, Endo-

crine effects. Induce differentiation/

growth/death. Balance is crucial!


H

P (ki)

! !

P (bi)

bi

k0

hi

BK

ki

b0

Helper cells make the system complex

B lymphocytes. Play as APC

. Make specific Abs. Need signal of chemical

messengers from Th

Tk lymphocytes. Induce death of dangerous cells. Need signal of chemical messengers from Th

Th lymphocytes. No effector functions. Receive first signal from APC. Release/Absorb regulatory agents (cytokines)

Cytokines. Auto-, Para-, Endo-

crine effects. Induce differentiation/

growth/death. Balance is crucial!


H

P (ki)

! !

P (bi)

bi

k0

hi

BK

ki

b0

Th lymphocytesEach clone hi = ±1, i = 1,...,HActual concentration ∼ Σi exp(hi)

B lymphocytesSelf-regulatory effectsEach clone bi =N(0,1), i = 1,...,B

Actual concentration ∼ Σi exp(bi)

Tk lymphocytesSelf-regulatory effectsEach clone ki =N(0,1), i = 1,...,K

Actual concentration ∼ Σi exp(ki)Cytokinesξiμ cytokine acting between hi and bμ (kμ), assumed symmetric

ξiμ = ± 1 for excitatory/inhibitory stimulation, P(ξiμ = +1) = P(ξiμ = -1) = 1/2{ξ} quenched and encodes proper “strategies”

3-partiteSG


Partition function:

Gauss-integrate

n-partite spin-glass Sum of n-1 independent neural networks

Helpers promote/suppress, via cytokines, the effectors branches

Cytokines directly connect helpers via Hebbian interaction, making them able to learn, store, retrieve patterns


1

2

3

4

1

2

3

1 2

3 4

-1

-1-1

+1

+1

+3


1

2

3

4

1

2

3

1 2

3 4

-1

-1-1

+1

+1

+3

1

2

3

4

Retrieval pattern ξ1 ⇔ Stimulation b1

1

2

3As a result of a learning process, cytokines

ensures the dynamic stability of certain helper configuration, corresponding to

stimulation of certain bi

Mattis magnetization for the μ-th state


Scaling for subset size (high load)α, γ standard hematological markers

As the amount of patterns increases, the network fails retrieval

CD4/CD8 <1 → HIV progression in AIDS “Inverted ratio” reduces lymphocyte activity and responsiveness to foreign antigens

System actually always subjected to external stimuli (viruses, bacteria, tumor cells) on effector branches → positive mean activity k0 [polyclonal expansion]

Random field → N(0,1), TDL

consistent with low-titer self-Abs


β noise levelΦ rescaled measure of mean activityα/γ lymphocyte ratios

Ordered work (organized effector activation) [∼ t]→ introduction of disorder within the system [∼ √t]

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

1/!

"

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

1/!

"

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

1/!

"

minimize RS free-energy w.r.t. order parameters

self-consistent equations for

Φ=0 Φ=0.5 Φ=1

SG

RetrievalPureState


Emergence of pathologies as retrieval failure

Abnormal lymphocyte accumulation (e.g. ALPS, leukemia)

→ severe autoimmune phenomena

Φ>>1 RF phase prevails against retrieval phase

HIV to overt AIDS[HIV infect and kill h]EBV [immortalize B cells]

→ populations imbalance yields reduced immune functions and responsiveness

SG emergence and retrieval failureSlow noise prevails

Free-radicals, cholesterol, damage accumulation, debris from lysis→ slow down in recognition processes and speed up other pathologies

PM emergence and retrieval failureFast noise prevails


Recovering the system as a dynamical process

Agreement between idiotypic and tripartite

approaches

Agreement between systemic and

dynamical approaches

Ornstein-Uhlenbeck process

Mean-Field approximationsassuming 〈J〉 exists strictly positive

Gaussian effectors activity with variance given by weighted connectivity


Getting close to biology


Cytokines{ξ} quenchedExcitatory/inhibitory/absent stimulation

P(ξiμ = +1) = P(ξiμ = -1) = (1-d)/2P(ξiμ = 0) = d

Finite degree of dilution h-h still FC Network more effectiveParallel retrieval occurs Critical parameters are correctedIntermediate dilutions SG phase emergesExtremely high dilution h-h network diluted

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-2 -1 0 1 2

Campi alla Barra T=0 Alfa=0.05

dil_0.0dil_0.1dil_0.2

dil_0.25dil_0.3

dil_0.35dil_0.4

dil_0.45dil_0.5dil_0.6dil_0.7dil_0.8

Field distributions

Getting close to biology


- Unifying picture including Burnet and Jerne theories

- Idiotypic network based on eliciting, complementarity-based couplings recovers basic facts in immunology

- Self/Non-self discrimination merges experimental results and ontogenesis features

- Learning and Memory needs complex interactions yielded by a “central control” provided by Th

- Errors ≠ Spurious states → Realize overlap of several strategies hence response hires several correlated lymphocytes

- Auto-immune diseases related to a few main sources of “noise”

- The tripartite system is consistent with idiotypic-network as well as dynamical approaches

Conclusions


Work in collaboration with:Adriano Barra, Francesco Guerra, Francesco Moauro

(Sapienza, Università di Roma)Raffaella Burioni, Aldo Di Biasio (Università di Parma)

Silvio Franz, Thiago Sabetta (Université Paris-Sud, Orsay)

List of related publications:

EA, A. Barra, A Hebbian approach to complex network generation, EPL (2011)

EA, A. Barra, F. Guerra, F. Moauro, A statistical mechanics on ALPS, J. Theor. Biol. (2011)

A. Barra, EA, Solving statistical mechanics on complex topologies trough neural network

techniques, JStat (2011)

EA, A. Barra, F. Guerra, K. Gervasi-Vidal, Associative Learning as origin of CFS, submitted (2011)

EA, A. Barra, Statistical mechanics of idiotypic immune networks, submitted (2011)

A. Di Biasio, EA, A. Barra, R. Burioni, Mean-field cooperativity in chemical kinetics, Theor.

Chem. Acc. (2011)

A. Barra, EA, A. Galluzzi, F. Guerra, F. Moauro, Parallel Processing Immune Networks, (2011)A. Barra, S. Franz, T. Sabetta, EA, Ontogenesis in B-cell and self-discrimination, (2011)

EA, C. Cioli, E. Guadagnini, Percolation on correlated random networks, PRE (2011)

A. Barra, EA, A Statistical Mechanics approach to autopoietic immune networks, JStat (2010)venerdì 23 settembre 2011


Low-Dose Tolerance c hik > wk

N

In “healthy state”, antigenic concentration to elicit response is

lower-bounded



Bell-shape response

0 500 1000 1500 2000−1

−0.5

0

0.5

1

1.5

2

t

mi,

mi,

x

Ab1

AAb1

Antigen

0 2000 4000 6000 8000

−0.5

0

0.5

1

tm

i,m

i,m

j,m

j,c

x

Ab1

AAb1

Ab2

AAb2

Antigen

First and second best matching antibody reacting with the antigen

Pathological example of a chronic response

Low-Dose Tolerance



Memory Effects

Antigen ! !1(Ab) ! !2(Ab) ! !3( ¯Ab)...Circuit of length l: " = {i1, i2, ..., il}Overall strength J! = l!1

!lk=1 Jik,ik+1 ! intrinsic robustness of "

Overall weight w! = l!1!l

k=1 wik ! influence of ext environment on "

Excited self-reinforced loops













100 102 1040

1000

2000

w!

10−6 100 1050

200

400

600

800

J!

l = 4, Self

l = 4, Non-Self

l = 3, Self

l = 3, Non-Self

The less diluted the network the smaller the region where memory effects may emerge

Region centered on <k> = O(1012) → L = O(102) consistent with experiments

Bell-shape responseLow-Dose Tolerance



Robustness under Ab correlation

Hypersomatic mutations in germinal centersEvolution not completely random Selective specification

➪ P(ξ=±1)= (1±a)/2a ∈ [-1,+1]

(ageing)

Correlation first induces fringes and then breaks down connectivity

- Low-Dose Tolerance- Bell-shape response- Memory Effects- Self-NonSelf Discrimination


N!10000, L!64, a!0.570000, "!0.700000.9098 nodi isolati.

N!10000, L!64, a!0.590000, "!0.700000.9472 nodi isolati.

N!10000, L!64, a!0.610000, "!0.700000.9787 nodi isolati.

N!10000, L!64, a!0.630000, "!0.700000.9896 nodi isolati.

Small (w.r.t. ER) frequency of short

loops in underpercolated regime

0.35 0.4 0.45 0.5 0.55 0.6 0.650

2000

4000

6000

8000

10000

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.650

2

4

6

8

10

a

#

−0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

2

0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5x 105

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5x 108

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.6510−5

100

105

1010

a

#

Scan di a. N=10000, L=64; !=0.7; 10 prove.

isolated nodes dimerstrimers3stars

triangles quadrangles trianglesquadrangles

0.35 0.4 0.45 0.5 0.55 0.6 0.650

2000

4000

6000

8000

10000

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.650

2

4

6

8

10

a

#

−0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

2

0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5x 105

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5x 108

a

#

0.35 0.4 0.45 0.5 0.55 0.6 0.6510−5

100

105

1010

a

#

Scan di a. N=10000, L=64; !=0.7; 10 prove.

isolated nodes dimerstrimers3stars

triangles quadrangles trianglesquadrangles


H

P (ki)

! !

P (bi)

bi

k0

hi

BK

ki

b0


Standard (Φ=0) phase diagramfor Hopfield model

P: paramagnetic phase, m=q=0SG: spin-glass phase, m=0, q≠0

F: pattern recall phase (pure states minimize f), m≠0, q≠0M: mixed phase (pure states are local but not global minima)

Tg: second order, TM and Tc: first order


0.2 0.4 0.6 0.8 1.0m

0.2

0.4

0.6

0.8

1.0

q

0.2 0.4 0.6 0.8 1.0m

0.2

0.4

0.6

0.8

1.0

q

0.2 0.4 0.6 0.8 1.0m

0.2

0.4

0.6

0.8

1.0

q

0.2 0.4 0.6 0.8 1.0m

0.2

0.4

0.6

0.8

1.0

q


the immune system from a statistical mechanics...

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