  The probability distribution of sequences with maximum entropy that satisfies
a given amino acid composition at each site and a given pairwise amino acid
frequency at each site pair is a Boltzmann distribution with $\exp(-\psi_N)$,
where the total interaction $\psi_N$ is represented as the sum of one body and
pairwise interactions. A protein folding theory based on the random energy
model (REM) indicates that the equilibrium ensemble of natural protein
sequences is a canonical ensemble characterized by $\exp(-\Delta G_{ND}/k_B
T_s)$ or by $\exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept
constant, meaning $\psi_N = \Delta G_{ND}/k_B T_s +$ constant, where $\Delta
G_{ND} \equiv G_N - G_D$, $G_N$ and $G_D$ are the native and denatured free
energies, and $T_s$ is the effective temperature of natural selection. Here, we
examine interaction changes ($\Delta \psi_N$) due to single nucleotide
nonsynonymous mutations, and have found that the variance of their $\Delta
\psi_N$ over all sites hardly depends on the $\psi_N$ of each homologous
sequence, indicating that the variance of $\Delta G_N (= k_B T_s \Delta
\psi_N)$ is nearly constant irrespective of protein families. As a result,
$T_s$ is estimated from the ratio of the variance of $\Delta \psi_N$ to that of
a reference protein, which is determined by a direct comparison between
$\Delta\Delta \psi_{ND} (\simeq \Delta \psi_N)$ and experimental $\Delta\Delta
G_{ND}$. Based on the REM, glass transition temperature $T_g$ and $\Delta
G_{ND}$ are estimated from $T_s$ and experimental melting temperatures ($T_m$)
for 14 protein domains. The estimates of $\Delta G_{ND}$ agree well with their
experimental values for 5 proteins, and those of $T_s$ and $T_g$ are all within
a reasonable range. This method is coarse-grained but much simpler in
estimating $T_s$, $T_g$ and $\Delta\Delta G_{ND}$ than previous methods.
