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When an antibody binds to unintended proteins (with highly similar epitopes) we speak of non-specificity. However, binding to closely related proteins may sometimes be unavoidable. Then the actual binding of the antibody may be specific, yet the antibody is deemed non-specific in relation to the intended target protein. Further diluting the antibody and optimizing blocking conditions will not work in such cases. So, the specificity of an antibody relies on the uniqueness of the protein part it binds to (epitope).
However, an antibody that binds to an epitope that is present in one or two other (closely related) proteins, may not be useless. It may still be useful in tissues or cell types where those cross-reacting proteins are not present. Or the scientist can take advantage in relating the intensities of bands representing the different proteins in Western blot.
Non-specific background can be reduced by further diluting the antibody. The reason is simple: by diluting the antibody you ask for higher affinity interactions. The lower affinity interactions (to remotely similar epitopes) will not last at lower antibody concentrations. Proper blocking conditions can also help preventing low affinity interactions.
Poor experimental conditions will incur random noise and this usually is not related to the primary antibody. Omitting blocking components, or dirty containers/contaminated buffers usually are the culprit. Especially in fluorescence, noise can be a big issue. There is danger for antibodies being dismissed prematurely because of not dealing with background and noise separately.
Secondary antibody derived signals
It is not only the primary antibody that can cause problems. The quality of the secondary antibody can be addressed by side-by-side comparison between a complete experiment with one without the primary antibody. Noise and background caused by the secondary antibody will be seen in this negative control.