Chapter 3
Logarithmic Growth of Protein Films Simon Alaeddine and Håkan Nygren
Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/bk-1995-0602.ch003
Department of Anatomy and Cell Biology, University of Göteborg, Medicinaregatan 5, S-413 90 Göteborg, Sweden
The kinetics of protein adsorption and antibody binding to surface -immobilised antigen was measured with off-null ellipsometry under non -diffusion limited conditions. An initial lag-phase was seen, followed by an accelerating reaction rate (auto catalysis). The reaction rate then decreased at a surface concentration far below monolayer coverage and a continuously decreasing rate of binding was seen. The kinetics can be described by a logistic law of limited growth function. The theoretical description of surface reactions with fractal kinetics is discussed. The reaction rate of macromolecular reactions at interfaces decreases logarithmically over long periods of time as shown for protein adsorption (7, 2 ) and antigen-antibody reactions (3,4). The general phenomenon of logarithmically decreasing reaction rates has been collectively named fractal kinetics ( 5 ) and has been demonstrated experimentally in a number of situations ( 6 ) . Spatial and/or energetic heterogeneity of the medium or non randomness of the reactant distribution in low dimensions have been suggested as mechanisms behind the phenomenon of fractal kinetics. Experimental studies of ferritin adsorption, a suitable model system with fractal kinetics, have revealed that the logarithmic growth of the protein film is preceded by an initial acceleration-phase of adsorption ( 2). The acceleration of the initial adsorption can be described by nucleation-andgrowth- like kinetics assuming attraction between adsorbed molecules and molecules in solution. The initial cooperative adsorption can be described theoretically by an exponential growth (7), which rapidly leads to depletion of reactants in the reaction zone making the reaction mass-transport limited. It has also been shown that ferritin clusters are restructured during adsorption with fractal kinetics. Orderly structured aggregates seen in previous phases of adsorption disappear and a more random distribution is seen. This process of rearrangement of dense clusters is not necessarily a continuous one but may take place through critical dissociation of orderly structured aggregates (2, 8 ). A n intellectual model of this phenomenon is the continuous build up and discontinuous fall down of sand on sand piles in an hour-glass. Theoretical models of such processes have been elaborated (9, 10 ) and may serve to explain the fractal kinetics of macromolecular reactions at surfaces. The use of such models have been suggested for intermolecular protein dynamics (77 ). The present study was undertaken in order to further describe the molecular mechanism behind the kinetics of macromolecular reactions at interfaces. 0097-6156/95/0602-0041$12.00/0 © 1995 American Chemical Society In Proteins at Interfaces II; Horbett, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.
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PROTEINS AT INTERFACES II
Theory 1. Nucleation. From previous studies with T E M it is known that there is a limited number of sites available for monomolecular binding. The number of nucleation sites differ between surfaces but is typically of the order of 10^-10^ molecules/cm (7, 12 ). It is not easy to measure the kinetics of this initial binding, but a probability for independent binding is relevant, at least for short incubation time (< 5 sec). Thus the initial nucleation rate can be written as (7): 2
no
Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/bk-1995-0602.ch003
^- = dt
R,k{t){N -N) max
(1)
where the function, k(t), corresponds to the sticking probability of independent binding at the nucleation sites (N ax)i N is the number of occupied nucleation sites; X is the frequency of collision with the nucleation sites; A u ^ is the mean activation energy per molecule; R is the molecular flux towards the surface (number of molecules striking l c m ^ of the surface per second) that is dependent on the boundary conditions of diffusion and the concentration of reactant in the bulk. The subscript "s" represents the concentration close to the surface. 2. Growth. The cluster can grow in an arbitrary shape, favouring the interaction energy, and may be described in different ways. One, two, and three-dimensional aggregates will grow if the mean interaction free energy, A|LL, decreases with the number of adsorbed molecules, S. Thus the interaction free energy of the molecules in the bulk phase can be expressed as (13) m
es
s
Hl=^
+^
L
(2)
where a . is a positive constant dependent on the strength of the intermolecular interactions and q is a number that depends only on the shape or the dimensions of the aggregates. For clusters adsorbed to the surface it is reasonable to assume that the interaction free energy is a superposition of the strength of the intermolecular interaction, a . , and the strength of the surface-molecules interaction, a m
m
m
m
s
& = £ +
B
s
^ - «
m
- , k T
m
(3)
The adsorbed protein film grows from initial nucleation sites, dependent on binding growth (14, 15) until critical clusters of various size and shape reach equilibrium with the bulk molecules. In the initial stages of adsorption, clusters of various sizes are in metastable equilibrium with adsorbed monomers. As these clusters grow, they deplete the surrounding region of adsorbed monomers so that further nucleation (or cluster formation) is not possible in this region (called a capture zone). Taking such a capture zone to be of radius r , one gets c
(4)
In Proteins at Interfaces II; Horbett, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.
3. ALAEDDINE & NYGREN
Logarithmic Growth of Protein Films
43
where D is the surface-diffusion coefficient of adsorbed monomers on the substrate, Ta is the mean free residence time before adsorbed monomer desorption. Hence the rate of addition of new molecules per cm^ per second is proportional to the number of occupied nucleation sites, N , the flux, the sticking probability, and the free residence time
(5)
where A|x i is the Gibbs free energy of activation for surface diffusion of adsorbed Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/bk-1995-0602.ch003
S(
monomers, Au
