  The soft condensed matter of biological organisms exhibits atomic motions
whose properties depend strongly on temperature and hydration conditions. Due
to the superposition of rapidly fluctuating alternative motions at both very
low temperatures (quantum effects) and very high temperatures (classical
Brownian motion regime), the dimension of an atomic ``path'' is in reality
different from unity. In the intermediate temperature regime and under
environmental conditions which sustain active biological functions, the fractal
dimension of the sets upon which atoms reside is an open question. Measured
values of the fractal dimension of the sets on which the Hydrogen atoms reside
within the Azurin protein macromolecule are reported. The distribution of
proton positions was measured employing thermal neutron elastic scattering from
Azurin protein targets. As the temperature was raised from low to intermediate
values, a previously known and biologically relevant dynamical transition was
verified for the Azurin protein only under hydrated conditions. The measured
fractal dimension of the geometrical sets on which protons reside in the
biologically relevant temperature regime is given by $D=0.65 \pm 0.1$. The
relationship between fractal dimensionality and biological function is
qualitatively discussed.
