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Antibody solutions are typically much more viscous than solutions of globular proteins at equivalent volume fraction. Here we propose that this is due to molecular entanglements that are caused by the elongated shape and intrinsic flexibility of antibody molecules. We present a simple theory in which the antibodies are modeled as linear polymers that can grow via reversible bonds between the antigen binding domains. This mechanism explains the observation that relatively subtle changes to the interparticle interaction can lead to large changes in the viscosity. The theory explains the presence of distinct power law regimes in the concentration dependence of the viscosity as well as the correlation between the viscosity and the charge on the variable domain in our antistreptavidin IgG1 model system.