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Optimal strategies in immunology

II. B memory cell production

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Summary

After a first encounter with most antigens, the immune system responds to subsequent encounters with a faster, more efficient and more strenuous antibody response. The memory of previous antigen contacts is carried by lymphocytes. Expanding on the model developed in Part I of this paper, we examine the optimal strategy available to the immune system for B memory cell production. We again find that the strategy should be of the bang-bang variety.

The model we consider assumes that antigen triggers a subpopulation of B-lymphocytes. These triggered lymphocytes can proliferate and secrete modest amounts of antibody, differentiate into non-dividing plasma cells which secrete large amounts of antibody, or differentiate into non-antibody secreting memory cells. Given injections of antigen at two widely spaced times we compute the strategy which minimizes a linear combination of the primary and secondary response times. We find that for all biologically reasonable parameter values the best strategies are ones in which memory cells are produced at the end of the primary response. Experimental results which bear on the actual strategies employed are discussed.

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Abbreviations

a :

age

A :

concentration of secreted antibody

A * :

concentration of antibody that must be secreted to eliminate antigen

b :

birth rate of large lymphocytes

b L :

net growth of large lymphocytes, b − μ L

c :

a constant

d :

rate of differentiation of large lymphocytes

d M :

rate of differentiation of large lymphocytes into memory cells

dP :

rate of differentiation of large lymphocytes into plasma cells

G(τ) :

defined by Eq. (A.38, Part I)

H :

Hamiltonian

I :

time interval

k :

antibody secretion rate

K :

terminal cost function, defined by Eq. (A.4)

L 0 :

concentration of large lymphocytes at the initiation of the primary response

L 20 :

concentration of large lymphocytes at the initiation of the secondary response that are derived from virgin (non-memory) small lymphocytes

M :

concentration of memory cells

M 20 :

concentration of non-stimulated memory cells at the initiation of the secondary response

M * :

concentration of memory cells at T 1, M(T 1)

P :

concentration of plasma cells

p :

factor weighting the importance of the secondary response

S :

target set

t :

time

t * :

switching time

T :

final time

T * :

optimal final time

u :

control vector, (u, v)

U :

control vector, (u, v, w)

u :

fraction of large lymphocytes that proliferate

v :

fraction of large lymphocytes that differentiate into plasma cells

v a :

reproductive value at age a

w :

fraction of large lymphocytes that differentiate into memory cells

x :

state vector, (A, L, P, M)t

α 2 :

scaled final antibody concentration, A *2 /k 2 L 20

γ :

ratio of plasma cell to large lymphocyte antibody secretion rates

η 2 :

A *2 k 2

θ :

function defining terminal manifold

Λ:

ratio of memory cells that become large lymphocytes at the initiation of the secondary response to the number of memory cells at the end of the primary response

Λ′:

Λ/L 20

λ:

costate vector

μ L :

death rate of large lymphocytes

μ P :

death rate of plasma cells

μ Ld :

net loss rate of large lymphocytes, λ L + d

Π:

Pareto set

σ :

switching function

τ :

time interval

1:

denotes primary response

2:

denotes secondary response

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Authors and Affiliations

  1. Theoretical Division, University of California, Los Alamos Scientific Laboratory, 87545, Los Alamos, New Mexico, USA

    Alan S. Perelson

  2. Department of Mechanical Engineering, University of California, 94720, Berkeley, California, USA

    Majdedin Mirmirani

  3. Division of Entomology, University of California, 94720, Berkeley, California, USA

    George F. Oster

Authors
  1. Alan S. Perelson
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  2. Majdedin Mirmirani
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  3. George F. Oster
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Additional information

Part I of this study is Perelson, A. S., Mirmirani, M. and Oster, G. F. ‘Optimal Strategies in Immunology. I. B-Cell Differentiation and Proliferation’, J. Math. Biol. 3, 325–367 (1976)

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Perelson, A.S., Mirmirani, M. & Oster, G.F. Optimal strategies in immunology. J. Math. Biology 5, 213–256 (1978). https://doi.org/10.1007/BF00276120

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  • DOI: https://doi.org/10.1007/BF00276120

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