Direct calculation of cryo EM and crystallographic model maps for real-space refinement

Real-space refinement of atomic models improves them by their fit to experimental scattering electrostatic potential maps in cryo electron microscopy and to electron density maps in crystallography. This procedure has a number of advantages in comparison with reciprocal-space refinement, is complementary to it in crystallographic studies and is the principal technique in cryo EM. An accurate real-space refinement of atomic models can be done by comparison of the model maps, calculated according to the respective theory, with the experimental ones, when the former mimic imperfections of the latter, mainly a limited resolution and an atomic disorder. Calculation of model maps as a sum of contributions of individual atoms means that these contributions - atomic images in the given map – should also be affected by the resolution and positional disorder. This blurs atomic images and surrounds their central peak by Fourier ripples. These ripples can be described by a specially designed function which allows combining both principal effects of map imperfection in an analytic way. The atomic images, at any resolution and with any value of the atomic displacement parameter, can be decomposed into a linear combination of such functions with the precalculated parameter values. As a consequence, each map value becomes an analytic function of atomic parameters including displacement parameter and a local resolution if required. Using the chain rule for the score function which compares the maps, such model results in analytic expressions for its partial derivatives with respect to all atomic parameters allowing an efficient real-space refinement. At the same time, for practical calculations atomic images are cut at some truncation distance, i.e ., include a limited number of ripples. This introduced in the model maps errors which are not present in the experimental maps. Due to oscillating behavior of the atomic images, the choice of the value of this truncation distance is not straightforward and discussed in this work. Synopsis A new method is suggested to calculate maps of a limited and eventually inhomogeneous resolution as an analytic function of all atomic parameters, including local resolution. ### Competing Interest Statement The authors have declared no competing interest.