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For example the dipeptide val-ser, between the amino acids valine and serine, does valine's amino group react or does it's carboxylic group with serine's respective group?

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    $\begingroup$ Proteins are named from N-terminus to C-terminus, so if the same rules apply for shorter peptide chains, then it is the carboxylic acid of valine that binds to the amino group of serine. $\endgroup$ – Tyberius Apr 13 at 19:42
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Under physiological conditions (aqueous solution, pH near neutral), amino acids don't react with each other directly, and the equilibrium lies on the side of peptide bond hydrolysis, not formation.

When amino acids react with each other, how do you know which one reacts using it's carboxylic group and which one with it's amine group?

In the cell, amino acids first react with ATP, then form an ester with RNA, and then their amino group forms a peptide bond with the growing amino acid chain. The last step is catalyzed by ribosomes that select the next amino acid to be added using messenger RNA as a template.

For example the dipeptide val-ser, between the amino acids valine and serine, does valine's amino group react or does it's carboxylic group with serine's respective group?

In your example, valine and serine could form either valyl-serine or seryl-valine. As Tyberius writes in the comments, peptides are written so that in each peptide link, the amino acid on the left of the link provides the carboxylic acid group and the one of the right of the link provides the amino group. The very first amino acid (the one at the amino terminus) carries an amino groups that has not undergone peptide bond formation, and the last amino acid (the one at the carboxy terminus) carries a carboxylate group that has not (yet) undergone peptide bond formation.

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Amino acids are organic compounds that have at least one amine group and one carboxylic group. They also contain a side chain specific to each amino acid. It is called an $\alpha$-amino acid if both amino and carboxylic groups of the amino acid are attached to the same carbon atom of its backbone (See the picture below for 21 proteinogenic $\alpha$-amino acids found in eukaryotes). If the amino and carboxylic groups are attached to separate adjacent carbon atoms of its backbone, then it is called a $\beta$-amino acid, and so on.

21 amino acids

Naturally, basic amino group and acid group can react to form amide bond. The formation of peptides is nothing more than the application of the amide synthesis reaction. By convention, the amide bond in the peptides should be made in the order that the amino acids are written. The amine end (N terminal) of an amino acid is always on the left, while the acid end (C terminal) is on the right. For example, consider the dipeptide, $\alpha$-Asp-Leu: This is a dipeptide that is the N-(L-$\alpha$-aspartyl) derivative of L-leucine (See LHS of following picture):

Dipeptides

Although L-aspartic acid has carboxylic group in its side chain, the free $\alpha$-amino group justify it's been a N-terminal. Based on the same reason, aspartic acid can be considered either $\alpha$- or $\beta$-amino acid. Therefore, if the peptide formation is involved with side chain carboxylic group instead (between same two aamino acids), then resultant dipeptide is called $\beta$-Asp-Leu (See RHS of the picture).

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