Titers do not depend on the number of variant antigens encountered in each B cell—FDC interaction.

(DOCX)</p

Residues making up the conserved region in the spike S2 domain.

Numbering is based on the SARS-CoV-2 spike protein (PDB ID: 6VXX). (DOCX)</p

Mean panel titers as a function of the number of mutations in panel antigens.

Mutations occur either in any variable residues (All variable residues) or in the same residues that are mutated in the sequences 1–6 (Seq1-6 mutated residues). Panel titers are calculated against panels of 100 antigens and 1000 antigens. Mutated residues take on a value of -4. B cells are assumed to encounter all antigens at a time on the FDC. (TIF)</p

Spike protein structures colored by conservation fraction with varying weights of the structural conservation fraction (F_struc) and biochemical conservation fraction (F_bio) using SARS-CoV-2 as a reference.

Green residues have conservation fractions above 0.8, blue residues have conservation fractions below 0.8 and are not in the RBD, and red residues have conservation fractions below 0.8 and are in the RBD. (TIF)</p

Collection and processing of GISAID SARS-CoV-2 sequences.

(DOCX)</p

Choice of number of escape mutations considered.

(DOCX)</p

Mean panel titers as a function of the number of mutations in panel antigens.

Mutations occur either in any variable residues (All variable residues) or in the same residues that are mutated in the sequences 1–6 (Seq1-6 mutated residues). Mutated residues take on values of -4. B cells encounter either all antigens at a time (All-antigen) or one antigen at a time (One-antigen) on the FDC. (TIF)</p

Scaffold proteins allow for signals to be distributed over many time scales.

<p>Variation of the survival probability with time scaled to characteristic time scales, <i>τ</i>. (A) <i>S</i>(<i>t</i>/<i>τ</i>) (on a semi-logarithmic scale) for different values of <i>ζ</i>. Values of <i>ζ</i> are given in the legend. Large deviations of exponential decay are observed near the optimal value of <i>ζ</i> (<i>ζ</i>  = 1, red). (B) Survival probabilities were fit to a stretched exponential function (). Values of stretching exponent <i>β</i> as a function of scaffold density <i>ζ</i> are shown. Two cases are considered: (1) kinases can be activated only while bound to a scaffold (red) and (2) kinases can be activated while in solution and bound to a scaffold (blue). <i>β</i> deviates most from a purely an exponential (<i>β</i> = 1) at the optimal value of scaffold density (<i>ζ</i>  = 1) in both cases.</p

Relation between first passage time statistics and signal duration.

<p>(A) Trajectories of <i>x</i>(<i>t</i>) on a semi-log plot. The abscissa represents time scaled by the characteristics time <i>τ</i>. Trajectories for three values of <i>β</i> are shown: <i>β</i> = 0.06 (blue), <i>β</i> = 0.8 (green), and <i>β</i> = 1.0 (red). <i>k_φ</i> = 5 for each curve. Smaller values of <i>β</i> result in larger values of <i>x</i>(<i>t</i>) at longer times. (B) Values of signal duration as a function of <i>β</i> for different choices of threshold, <i>T</i> (defined in text); values of <i>T</i> are provided in the legend. For small values of <i>T</i>, <i>β</i><1 (i.e., the presence of scaffolds) markedly increases signal duration.</p

Power spectra for kinase activation at high, low, and optimal scaffold concentrations.

<p>Plots of where , are considered. Three cases are considered: low concentration (<i>ζ</i> = 0.005, <i>τ</i> = 1.9×10⁵<i>mcsteps</i>, <i>β</i> = 1.03), high concentration (<i>ζ</i> = 3.5, <i>τ</i> = 2.2×10⁷<i>mcsteps</i>, <i>β</i> = 0.97), and optimal concentration (<i>ζ</i> = 1.0, <i>τ</i> = 3.0×10⁶<i>mcsteps</i>, <i>β</i> = 0.60). On the <i>x</i>-axis, frequency is reported in units that are scaled to the characteristic time for the <i>ζ</i> = 1.0 case, <i>τ^opt</i> = 3.0×10⁶<i>mcsteps</i>. The <i>y</i>-axis contains values of <i>P</i>(<i>ωτ^opt</i>)/(<i>τ^opt</i>)².</p