Viruses and bacteria commonly exhibit spatial repetition of surface molecules that directly interface with the host immune system. However the complex interaction of patterned surfaces with immune molecules containing multiple binding domains is poorly understood. We developed a pipeline for constructing mechanistic models of antibody interactions with patterned antigen substrates. Our framework relies on immobilized DNA origami nanostructures decorated with precisely placed antigens. The results revealed that antigen spacing is a spatial control parameter that can be tuned to influence antibody residence time and migration speed. The model predicts that gradients in antigen spacing can drive persistent, directed antibody migration in the direction of more stable spacing. These results depict antibody-antigen interactions as a computational system wherein antigen geometry constrains and potentially directs antibody movement. We propose that this form of molecular programmability could be exploited during co-evolution of pathogens and immune systems or in the design of molecular machines.