A. Two step binding scheme for a polyreactive Fab binding to an epitope.

<p>The epitope is exposed transiently for a mean time 1/ before virion and target cell membranes fuse and infection becomes irreversible. Only the transmembrane domain and the MPER on the viral membrane are pictured. The remainder of gp41, which extends into the target cell membrane, is not shown. The first step, binding from solution to the MPER on the virion surface is described by rate constants and . The second step, an induced conformational change resulting in a long-lived Fab-MPER complex, is described by intramolecular rate constants and . The model assumes that if an Fab remains bound to the MPER after the conformational change for a sufficient time () the epitope is disabled. B. In the encounter model the Fab binds first to lipids on the membrane with rate constants and , then diffuses to the MPER and reacts with surface rate constants and . Because binding to the lipid is weak and rapidly reversible we show in <b>SI1</b> of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.s001" target="_blank">Text S1</a> that for this model and . Here is the lipid equilibrium constant and is the surface concentration of lipid binding sites.</p

An example of how the probability of disabling a bond when there are bonds present might depend on .

<p>A. Because of the repulsive forces between membranes, when a single bond bridges two membranes it will be stretched further than a bond that is one of many holding the membranes together. B. In this example the probability of breaking a bond in the presence of a large number of other bonds , (closed circles), <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e259" target="_blank">Eq. (14)</a>, and (open circles) was calculated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e268" target="_blank">Eq. (15)</a> with and . The values obtained for are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi-1003431-t003" target="_blank">Table 3</a>.</p

Fit of the model to the four neutralization experiments shown in Figure 5.

<p>A. A global nonlinear least squares fit was performed, fitting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e300" target="_blank">Eq. (18)</a> to the four experiments with five free parameters, four and . The best fit values obtained were: , and for the values in g/ml, 4E10 IgG = , 4E10 Fab = , 2F5 IgG = , and 2F5 Fab = . The neutralization data is plotted versus using the determined values. The solid curve is , given by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e300" target="_blank">Eq. (18)</a>, with and the best fit values for the four . In B and C, was fixed at and respectively and only the four values were varied. The neutralization data is plotted versus using the determined values for the fixed values of .</p

Estimating the Probability of Polyreactive Antibodies 4E10 and 2F5 Disabling a gp41 Trimer after T Cell-HIV Adhesion

<div><p>A few broadly neutralizing antibodies, isolated from HIV-1 infected individuals, recognize epitopes in the membrane proximal external region (MPER) of gp41 that are transiently exposed during viral entry. The best characterized, 4E10 and 2F5, are polyreactive, binding to the viral membrane and their epitopes in the MPER. We present a model to calculate, for any antibody concentration, the probability that during the pre-hairpin intermediate, the transient period when the epitopes are first exposed, a bound antibody will disable a trivalent gp41 before fusion is complete. When 4E10 or 2F5 bind to the MPER, a conformational change is induced that results in a stably bound complex. The model predicts that for these antibodies to be effective at neutralization, the time to disable an epitope must be shorter than the time the antibody remains bound in this conformation, about five minutes or less for 4E10 and 2F5. We investigate the role of avidity in neutralization and show that 2F5 IgG, but not 4E10, is much more effective at neutralization than its Fab fragment. We attribute this to 2F5 interacting more stably than 4E10 with the viral membrane. We use the model to elucidate the parameters that determine the ability of these antibodies to disable epitopes and propose an extension of the model to analyze neutralization data. The extended model predicts the dependencies of for neutralization on the rate constants that characterize antibody binding, the rate of fusion of gp41, and the number of gp41 bridging the virus and target cell at the start of the pre-hairpin intermediate. Analysis of neutralization experiments indicate that only a small number of gp41 bridges must be disabled to prevent fusion. However, the model cannot determine the exact number from neutralization experiments alone.</p></div

The probability of disabling an epitope, , versus the rate at which membrane fusion occurs, , for different values of the disabling rate, .

<p>The probability was calculated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e136" target="_blank">Eqs. (7)</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e144" target="_blank">(8)</a> for two different 4E10 Fab concentrations, M in A and M in B, using the parameter values given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi-1003431-t002" target="_blank">Table 2</a>. The values of in used to generate the various curves are given in the figures. The curves for are indistinguishable. The top curves (solid black) in A and B each correspond to three curves generated for , and .</p

The values of (Eq. (14)), (Eq.(15)), and for the example in Figure 6.

<p>The values of (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e259" target="_blank">Eq. (14)</a>), (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e268" target="_blank">Eq.(15)</a>), and for the example in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi-1003431-g006" target="_blank">Figure 6</a>.</p

Binding scheme for a polyreactive antibody.

<p>A. The binding reactions when the antibody binding sites cannot bind directly to lipids in the membrane. B. The binding reactions when the antibody first binds to lipid in the membrane and then interacts with the MPER. is the surface equilibrium cross-linking constant for an antibody with one site free and one site bound to the membrane or the MPER binding to the membrane to form a bivalent attachment.</p

Neutralization of HIV-1 pseudovirus by Fab and IgG 2F5 and 4E10.

<p>The fraction neutralized is plotted against the concentration of Fab sites. The inserts in A, B and C are the calculated values of from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e136" target="_blank">Eq. (7)</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi.1003431.e162" target="_blank">(10)</a> versus the concentration of Fab sites for and the parameters given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003431#pcbi-1003431-t002" target="_blank">Table 2</a>. Neutralization by A. 2F5 Fab (red circles) and two preparations of 2F5 IgG (solid squares and open circles), B. 4E10 Fab (blue squares) and 4E10 IgG (green circles), and C. 2F5 IgG from panel A and 4E10 IgG from panel B.</p

Definition of the parameters of the model.

<p>Definition of the parameters of the model.</p

Recent Advances in Mathematical Programming with Semi-continuous Variables and Cardinality Constraint

Abstract Mathematical programming problems with semi-continuous variables and cardinality constraint have many applications, including production planning, portfolio selection, compressed sensing and subset selection in regression. This class of problems can be modeled as mixed-integer programs with special structures and are in general NP-hard. In the past few years, based on new reformulations, approximation and relaxation techniques, promising exact and approximate methods have been developed. We survey in this paper these recent developments for this challenging class of mathematical programming problems.