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Eukaryotic cells transmit information by signaling through complex networks of interacting proteins. Here we develop a theoretical and computational framework that relates the biophysics of protein-protein interactions (PPIs) within a signaling network to its information processing properties. To do so, we generalize statistical physics-inspired models for protein binding to account for interactions that depend on post-translational state (e.g. phosphorylation). By combining these models with information-theoretic methods, we find that PPIs are a key determinant of information transmission within a signaling network, with weak interactions giving rise to "noise" that diminishes information transmission. While noise can be mitigated by increasing interaction strength, the accompanying increase in transmission comes at the expense of a slower dynamical response. This suggests that the biophysics of signaling protein interactions give rise to a fundamental "speed-information" trade-off. Surprisingly, we find that cross-talk between pathways in complex signaling networks do not significantly alter information capacity--an observation that may partially explain the promiscuity and ubiquity of weak PPIs in heavily interconnected networks. We conclude by showing how our framework can be used to design synthetic biochemical networks that maximize information transmission, a procedure we dub "InfoMax" design., Comment: 13+10 pages, 5+9 figures, Supplemental Information included