Physical Chemistry of Protein Crystallization

EMBL Practical Course on Protein Expression, Purification and Crystallization

August 14^th-20^th, 2000 EMBL Outstation Hamburg, Germany

Bernhard Rupp

University of California, LLNL-BBRP, Livermore, CA 94551

Institut für Theoretische Chemie und Strukturbiologie der Universtät Wien, A 1090 Wien

© 2000 Bernhard Rupp

Introduction. 2

Thermodynamic stability. 2

Ideal systems. 2

Real systems and non-ideality. 4

Virial expansions. 6

Thermodynamic stability in multi-component systems. 8

Equilibrium and stability in G/x space. 10

Binary G/x diagrams. 11

Phase diagrams. 12

Binary phase diagrams. 12

Ternary phase diagrams. 13

Pathways in different crystallization methods. 14

Summary. 15

References. 16

Introduction

Protein crystallization can be viewed as a special case of phase separation in a thermodynamically non-ideal mixture controlled by kinetic parameters: The protein has to separate from aqueous solution and should form a distinct and hopefully, well ordered crystalline solid phase. We will therefore discuss the thermodynamics of non-ideal phase equilibria. Non-ideality leads to thermodynamic excess properties, which manifest themselves in fugacity, activity, and mixing enthalpies. Virial expansions relate non-ideality directly to coefficients interpretable as a measure for the interactions between protein and solvent. We will derive bimodal (solubility) and spinodal (decomposition) curves and construct the corresponding phase diagrams. Properties derived from thermodynamic parameters, including virial coefficients, are invariant for a given system and determine only the range of possible crystallization, without any predictive quality to its actual occurrence. The realization of a thermodynamically possible scenario is governed by kinetic parameters such as nucleation and crystal growth mechanisms, which are at the current state of the art unpredictable, but can be investigated and analyzed a posteriori to derive models and empirical parameters for crystallization processes.

Thermodynamic stability

Given that a protein solution represents a multi-component thermodynamic system, we need to investigate which criteria determine stability and what separates stable from non- or meta-stable regions in a given system. First, we will consider the most simple case, an ideal versus a non-ideal gas, to introduce basic stability criteria and virial expansions. Subsequently we will discuss the general concept of thermodynamic stability using the second Legendre transform G(T,P,n) of the generalized Gibbs fundamental equation to derive stability under variation of composition. In the final step, we shall use the partial molar Gibbs energy (chemical potential) to construct the phase diagrams as used to describe crystallization pathways in protein-solvent systems. Fortunately, although comprehensive, these basics are much less difficult to understand than it initially might appear.

Ideal systems

The simplest case of a thermodynamic system is a gas of ideal (dimensionless and non-interacting) particles in a cylinder with a reversible piston at a fixed temperature T. Varying pressure P exerted on the piston will reversibly compress or expand the gas to a certain volume V.

From the ideal gas law follows that

(1)

With T and n (the number of moles of gas) constant (R is already the gas constant, with a fixed numeric value of 8.3145 JK^-1M^-1) we see that the variation of volume vs. pressure is a simple hyperbolic function