Dr.Abel MORENO CARCAMO Instituto de Química, UNAM.

E-mail address: carcamo@sunam.mx

When? Understanding the macromolecular scale in

time for crystal growth phenomena

It is typical for crystal growth that many different length scales occur simultaneously: 0.01 ns for molecular conformations, ns for surface structure and defect displacements, µs for surface step displacement, ms for growth on one atomic layer, seconds for hydrodynamic transport, and minutes for homogenous nucleation.

Why are DLS & SLS needed?

For problems of homogeneity, nucleation, protein-protein interactions, precrystallization conditions, and recently to predict the induction time for growing protein crystals.

How?

Growth from solutions is usually predicted by hydrodynamic properties using dynamic and static light scattering methods.

Dynamic Light Scattering

Photon correlation spectroscopy (PCS) also known as dynamic light scattering (DLS) is concerned with the investigation of correlation of photons.

The objective of PCS is to find any peculiar properties of the scattered signal which can be used to characterize and describe the seemingly random “noise” of the signal, and the correlation curve is used to achieve this objective.

What is needed from the experimental viewpoint?

How does DLS work?

In principle, dynamic light scattering (DLS) can be used to study any phenomenon that gives rise to an intensity fluctuation in the light that has been scattered from a scattering source:

a)  Protein monomers

b)  Protein aggregates

c)  Micro-crystals

d)  Precipitating agents

e)  Additives

f)  Even dust particles.

Each scattering source will scatter light with a characteristic intensity (static property), which we will give the static light scattering data and a characteristic scattered intensity fluctuation (dynamic property) giving us the dynamic light scattering data.

Light scattering and crystal growth

Using dynamic light scattering to screen and improve protein crystal growth procedures, one does not need to employ overly sophisticated data reduction schemes. From the crystal growth point of view, one would like to be able to:

a)  detect the formation of the small protein aggregates which allows nucleation and growth to be monitored,

b)  estimate the mean value for the diffusion coefficient and hence the particle size of the protein molecules under various solutions conditions, and

c)  characterize aggregates as protocrystals or protoprecipitates

What kind of solid phase is obtained?

Early in the evolution of a supersaturated solution of macromolecules, monomers assemble to form aggregates, which lead either to crystals or precipitates. In an obvious terminology it is convenient to refer to the former as CRAGGS (crystalline aggregates) and the latter as PRAGGS (precipitating aggregates).

DLS & SLS for PROTEINS

The reported uses of DLS & SLS for the specific purpose of studying crystallizing protein solutions have generally been for:

a)  Characterize of the thermodynamic properties of the solutions where aggregates are forming

b)  Monitoring the nucleation kinetics for control purpose.

Reference:

W.W. Wilson. Methods: A companion to methods in enzymology, 1, No.1 (1990) 110-117.

!Figure 2: Schematic representation of the different nucleation mechanisms, starting from (a) a supersaturated solution to (b) a crystal. Reprinted with permission from Erdemir et al.21. Copyright 2009 American Chemical Society.

NUCLEATION: THE BIRTH OF CRYSTALS

Crystalliza*on  process  and  nuclea*on  Energe&cs  of  the  whole  crystalliza&on  process:          Δµ  =  µs    -­‐    µc        …  (1)    were µs and µc are the chemical potentials of a molecule in the solution and in the bulk of the crystal phase, respectively. When  Δµ  >  0,  the  solu&on  is  supersaturated,  and  only  then  is  nuclea&on  and/or  crystals  growth  possible.  The  solu&on  is,  respec&vely,  saturated  or  under-­‐saturated  when  µs    -­‐    µc  =  0  or  Δµ  <  0.    Δµ  =  kT  ln  β        …  (2)                  on  this  equa&on  β  =  C  /Ce  =  supersatura&on.    where k is the Boltzmann constant, T is the absolute temperature. The β the super-saturation, i.e. the ratio of actual solute concentration (more precisely the activity) divided by the concentration (or activity) at the equilibrium.  The  work  W  (J)  to  form  a  cluster  of  n  =  1,  2,  3…  molecules  can  be  obtained  by  thermodynamic  considera&ons      W(n)  =  -­‐nΔµ  +  Φ(n)        …  (3)  Φ  (J)  represents  the  effec&ve  excess  energy  of  the  cluster,  while  Δµ  is  given  by  

Equa&on  2  

According   to   the   classical   nuclea&on   theory,  Φ(n)   is  merely   equal   to  the   cluster   total   surface   energy   γAc(n)  where   γ   (J  m-­‐2)   is   the   specific  surface  energy  of  the  cluster/solu&on  interface,      Ac(n)  =  (36πνo2)1/3  n2/3  is  the  area  of  the  cluster  surface.      In   this   case,   νo   (m3)   is   the   volume   occupied   by   a   molecule   in   the  cluster.      W(n)  =  -­‐n  k  T  ln  β  +  (36  π  νo2)1/3  γ  n2/3      …  (4)    γ  =  Bλ/νo2/3  …    (5)    In  this  equa&on,  B  is  a  numerical  factor  (ranging  from  ≈  0.2  to  0.6),  this  value  is  approximately  equal  to  0.514  for  spherical  clusters,  and  λ  (J)  is  the  molecular  heat  of  dissolu&on  .  For  the  case  of  proteins  and  other  biological   macromolecules   this   value   must   be   adjusted   to   0.002   for  quasi-­‐spherical  par&cles.  

This  implies  that  γ  is  propor&onal  to  the  ln  Ce  and  indeed  is  the  case,  since  λ  and  the  solubility  Ce  are  related  by  Ce  =  (1/νo)exp(-­‐λ/kT)  as  described  by  Moelwyn-­‐Hughes  .    γ  =  (B  k  T  /  νo2/3)  ln  [1  /  νo  Ce  (T)]      …(6)  

So  that:      λ  =  -­‐kT  ln  (Ce  νo)    …(7)    Finally  for  the  surface  energy,  we  can  get  a  whole  equa&on  as  follows:    γ  =  B  [  -­‐kT  ln  (Ce  νo)]/νo2/3      …  (8)    For  the  crystalliza&on  process  we  have  the  following  equa&on:        ΔG  =  -­‐  [(4/3  π  r3)  /  Ω]  k  T  ln  β  +  4  π  r2  γ        …  (9)    The   supersatura&on   is   defined   by  β   =   C/Ce   or      α=   [(C-­‐Ce)/Ce].   In   solu&on   some  parameters  and  surface  energy  can  be  derived   from  Juarez-­‐Mar&nez  and  Moreno  Equa&on:      2  γ  vo  /  r  =  k  T  ln  β    +  A2  k  T  Mr  Ce  (β-­‐1)      ...  (10)  

Substance Mr (kDa) γ (erg/cm2) H2O content (%) Reference

DMSO 78 x 10-3 34.9 0 77

Benzene 78 x 10-3 28.5 0 77

Glycerol 92 x 10-3 26.1 0 77

Insulin 7.3 1.5 65 78

Lysozyme 14.5 1.09 27 79

Chymotrypsinogen 27 0.81 51 79

Thermolysine 34 0.93 55 79

Concanavalin A 102 0.44 44 79

Apo-ferritin 443 0.027 62 80

STMV 1500 0.018 56 79

* kDa in MW is generally used only for proteins.

Reference: Sanchez-Puig et al. Protein and Peptide Letters. Vol 19, June 2012.

The Physicochemical parameters to understand the crystallization process

Reference: Sanchez-Puig et al. Protein and Peptide Letters. Vol 19, June 2012.