TY - JOUR

T1 - Retention of antigen on follicular dendritic cells and B lymphocytes through complement-mediated multivalent ligand-receptor interactions

T2 - Theory and application to HIV treatment

AU - Hlavacek, William S.

AU - Percus, Jerome K.

AU - Percus, Ora E.

AU - Perelson, Alan S.

AU - Wofsy, Carla

N1 - Funding Information:
We thank Byron Goldstein for helpful discussions. This work was supported by the Department of Energy under contract W-7405-ENG-36, grants RR06555 and AI28433 from the National Institutes of Health, and grant MCB9723897 from the National Science Foundation. Appendix A Proof and Theorem 1 We present the proof for the special case k r = k − x . Extension to the case k r ≠ k − x is straightforward. Taking the condition k r = k − x into consideration, we can rewrite Eqs. (4) and (5) as follows: (A.1) t 1 = 1 k −x (1+α) n −1 nα , (A.2) t n = 1 k −x ∑ n j=1 ∑ n−j k=0 (n−j)!(j−1)! (n−j−k)!(j+k)! α k . Where α = K x R T (recall that K x = k + x / k − x ). To obtain a bound for t n that can be compared with the expression for t 1 ( Eq. (A.1) ), we will use relations between combinatoric terms and definite integrals. We will go from the double sum in Eq. (A.2) to a double integral and finally to a simple single sum ( Eq. (A.10) ). From Eq. (A.2) , (A.3) k −x t n =∑ n j=1 ∑ n−j k=0 n−j k (j−1)!k! (j+k)! α k . It is straightforward to check (in a table of integrals or using integration by parts) that (A.4) (j−1)!k! (j+k)! =∫ 1 0 (1−x) j−1 x k d x. Then from Eq. (A.3) , (A.5) k −x t n =∑ n j=1 ∑ n−j k=0 n−j k ∫ 1 0 (1−x) j−1 x k α k d x=∑ n j=1 ∫ 1 0 (1−x) j−1 ∑ n−j k=0 n−j k (αx) k d x. The bracketed expression in Eq. (A.5) is the binomial expansion of (1+ αx ) n − j . Therefore (A.6) k −x t n =∑ n j=1 ∫ 1 0 (1−x) j−1 (1+αx) n−j d x=∫ 1 0 (1+αx) n−1 ∑ n j=1 1−x 1+αx j−1 d x. Summing the geometric series in Eq. (A.6) , gives (A.7) k −x t n =∫ 1 0 (1+αx) n−1 1−x 1+αx n −1 1−x 1+αx −1 d x=∫ 1 0 (1+αx) n −(1−x) n x(1+α) d x. The integrand in Eq. (A.7) can be written as a definite integral, so that (A.8) k −x t n =∫ 1 0 ∫ 1 0 n yx(1+α)+1−x n−1 d y d x. A binomial expansion, followed by integration with respect to y yields (A.9) k −x t n =n∫ 1 0 ∫ 1 0 ∑ n−1 k=0 n−1 k (yx(1+α)) k (1−x) n−1−k d y d x=n∑ n−1 k=0 1 k+1 n−1 k (1+α) k ∫ 1 0 x k (1−x) n−1−k d x. Applying Eq. (A.4) , (A.10) k −x t n =n∑ n−1 k=0 1 k+1 n−1 k (1+α) k (n−1−k)!k! n! =∑ n−1 k=0 (1+α) k k+1 =∑ n−1 j=0 (1+α) n−1−j n−j . Eq. (A.10) will be used to obtain a bound on k − x t n that can be compared with k − x t 1 . The bound depends on the following inequality, which can be checked directly. For 0⩽ j ⩽ n −1, (A.11) 1 n−j ⩽ 1 n + j(j+1) n 2 . From Eqs. (A.10) and (A.11) , (A.12) k −x t n ⩽ (1+α) n−1 n ∑ n−1 j=0 (1+α) −j + (1+α) n+1 n 2 ∑ n−1 j=0 (j+1)j(1+α) −j−2 . The first summation term in Eq. (A.12) is the sum a geometric series; the second summation term is the second derivative of the sum of the same geometric series. The inequality can be rewritten in the following form: (A.13) k −x t n ⩽ (1+α) n−1 n 1−(1+α) −n 1−(1+α) −1 + (1+α) n+1 n 2 d 2 d α 2 1−(1+α) −n 1−(1+α) −1 = (1+α) n nα 1−(1+α) −n + (1+α) n+1 n 2 d 2 d α 2 1+α−(1+α) −(n−1) α . The second derivative in Eq. (A.13) is the sum of 2/ α 3 and negative terms, which we will neglect. Thus, (A.14) k −x t n ⩽ (1+α) n −1 nα + 2(1+α) n+1 n 2 α 3 . Combining Eqs. (A.1) and (A.14) , (A.15) n→∞. t n t 1 ⩽1+ 2(1+α) nα 2 1−(1+α) −n =1+ O (1/n) as

PY - 2002

Y1 - 2002

N2 - In HIV-infected patients, large quantities of HIV are associated with follicular dendritic cells (FDCs) in lymphoid tissue. During antiretroviral therapy, most of this virus disappears after six months of treatment, suggesting that FDC-associated virus has little influence on the eventual outcome of long-term therapy. However, a recent theoretical study using a stochastic model for the interaction of HIV with FDCs indicated that some virus may be retained on FDCs for years, where it can potentially reignite infection if treatment is interrupted. In that study, an approximate expression was used to estimate the time an individual virion remains on FDCs during therapy. Here, we determine the conditions under which this approximation is valid, and we develop expressions for the time a virion spends in any bound state and for the effect of rebinding on retention. We find that rebinding, which is influenced by diffusion, may play a major role in retention of HIV on FDCs. We also consider the possibility that HIV is retained on B cells during therapy, which like FDCs also interact with HIV. We find that virus associated with B cells is unlikely to persist during therapy.

AB - In HIV-infected patients, large quantities of HIV are associated with follicular dendritic cells (FDCs) in lymphoid tissue. During antiretroviral therapy, most of this virus disappears after six months of treatment, suggesting that FDC-associated virus has little influence on the eventual outcome of long-term therapy. However, a recent theoretical study using a stochastic model for the interaction of HIV with FDCs indicated that some virus may be retained on FDCs for years, where it can potentially reignite infection if treatment is interrupted. In that study, an approximate expression was used to estimate the time an individual virion remains on FDCs during therapy. Here, we determine the conditions under which this approximation is valid, and we develop expressions for the time a virion spends in any bound state and for the effect of rebinding on retention. We find that rebinding, which is influenced by diffusion, may play a major role in retention of HIV on FDCs. We also consider the possibility that HIV is retained on B cells during therapy, which like FDCs also interact with HIV. We find that virus associated with B cells is unlikely to persist during therapy.

KW - B lymphocyte

KW - Birth-death Markov chain

KW - Follicular dendritic cell

KW - Human immunodeficiency virus type 1

KW - Immune memory

KW - Multivalent ligand-receptor binding

UR - http://www.scopus.com/inward/record.url?scp=0036231339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036231339&partnerID=8YFLogxK

U2 - 10.1016/S0025-5564(02)00091-3

DO - 10.1016/S0025-5564(02)00091-3

M3 - Article

C2 - 11916508

AN - SCOPUS:0036231339

VL - 176

SP - 185

EP - 202

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 2

ER -