Understanding the mechanisms of molecular spontaneous self-assembly is essential for the ability to predict structures of folded proteins and other complex biological structures. Precise control of material crystal structure and, therefore, its mechanical properties is another field where detailed knowledge of self-assembly is of principal interest.Our current knowledge of the mechanisms and kinetics of such processes is still limited. In this work the spontaneous growth of ice (freezing) is studied by molecular dynamics simulations in the isoconfigurational ensemble at three different temperatures below the melting point. Ice is a molecular crystal where water molecules are held in place by hydrogen bonding, an interaction similar to interactions in biological systems. This similarity and relative simplicity of this system, at the same time, make it a perfect subject for uncovering details of ordering and disordering processes during self-organization of matter.It is shown that specific structures determine local thermodynamics at a growing interface and directly influence kinetics of growth at a time scale of 1-2 ns due to fluctuations. The structural effect on the growth behaviour can be characterized in terms of relative growth propensities.The topology of the initial interfaces is obtained using a structural order parameter and compared with the observed growth behaviour. Critical interfacial features specific to the observed growth patterns are identified in some cases. The work clearly indicates that local structure determines, to a large degree, the tendency of an interface to grow or melt.This structural effect upon the ordering kinetics should be a universal behaviour and can be expected in more complex biologically relevant ordering processes.