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I know that at physiological pH, Arginine will have its $\ce{=NH}$ protonated to $\ce{=NH2+}$. I was wondering if this $\ce{=NH2+}$ can still form a hydrogen bond by being a donor? And does the positive charge on Nitrogen affect the hydrogen bonding's strength? My confusion stems from the definition of hydrogen bonding, which says that hydrogen's electron cloud needs to be withdrawn by a more electronegative atom like $\ce{F, O, N}$ to be partially positive enough to participate. Therefore, I believe that the positively charged N will withdraw electrons from hydrogens more strongly, leading to more partially positively charged hydrogen and eventually stronger H-bonding with another molecule. Could someone please verify this for me?

Moreover, I have seen $\ce{=O}$ and $\ce{=N}$ forming hydrogen bonding by being an acceptor. However, I tried search but did not find any example of $\ce{=NH3+}$ or $\ce{=NH2R+}$ or NHR2+ forming hydrogen bonding as a donor. So can a Nitrogen with a double bond still participate in hydrogen bonding as a donor?

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Yes, the arginine side chain is an excellent hydrogen bond donor. Charged N-H groups are even better hydrogen bond donors than the corresponding neutral N-H groups. Here is an article that gives examples of arginine side chains in hydrogen bonds. It showcases bidendate hydrogen bonds (i.e. two hydrogen bonds from the arginine side chain) with carboxylates and with nitrogenous bases of nucleic acids.

Specifically for proteins, if the hydrogen bond donor is positively charged (e.g. Arg+) and the hydrogen bond acceptor is negatively charged (e.g. Glu-), you have a combination of ionic and hydrogen bond interaction, which is particularly strong. It is often referred to as a salt bridge, such as in this example. If you take a close look at the arginine side chain, you will see that it is protonated (i.e. positively charged) and interacts with two negatively charged carboxylate groups.

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