VOLUME 83, NUMBER 22 PHYSICAL REVIEW LETTERS 29 NOVEMBER 1999 Depletion Interactions in the Protein Limit: Effects of Polymer Density Fluctuations Amit M. Kulkarni, 1 Avik P. Chatterjee, 2, * Kenneth S. Schweizer, 1,2 and Charles F. Zukoski 1 1 Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801 2 Departments of Materials Science & Engineering and Chemistry, University of Illinois, Urbana, Illinois 61801 (Received 8 March 1999) We report the first observations of nonmonotonic changes in the second virial coefficients of protein solutions, B 2 , as the concentration of nonadsorbing polymer is increased. The observed minimum in B 2 cannot be predicted from standard depletion interaction energy models and is closely associated with proximity to the lower critical solution temperature of the polymer solution. The location, depth, and molecular weight dependence of the minima are captured by the thermal polymer reference interaction site model for depletion interactions, where the polymer mesh size is a function of temperature. PACS numbers: 61.25.Hq, 64.75. + g, 82.70.Dd, 87.15.Kg The addition of nonadsorbing macromolecules to a stable particle suspension results in a polymer-mediated “depletion attraction” between particle pairs. Qualita- tively, this induced interaction arises from an unbalanced osmotic pressure due to the excluded volume driven ex- pulsion of polymers from the region between the particles. This phenomenon has significant scientific and technologi- cal consequences in diverse fields and has seen exten- sive investigation [1]. Most studies consider the “colloid limit” where the particle radius R greatly exceeds a statis- tically averaged measure of polymer size, the radius of gy- ration R g . The model proposed by Asakura and Oosawa (AO) was the first to describe the depletion effect in the colloid limit [2]. This approach approximates a flexible polymer chain as a rigid sphere of radius R g , ignores polymer-polymer interactions (dilute macromolecule, or “ideal solution” limit), and treats particle-particle and particle-polymer interactions as hard core repulsions. The AO model has been employed to interpret direct mea- surements of depletion forces [3–5], and as an effective pair decomposable potential, Ur , in a one-component de- scription of the properties of nondilute colloidal suspen- sions [1]. Specifically, the AO depletion potential is [2] Ur   2 4 3 p d 3 n p kT ∑ 1 2 3r 4d 1 r 3 16d 3 ∏ , 2R # r # 2d (1) and zero ` for r . 2d r , 2R, where d  R 1 R g , and n p is the polymer number density. The AO potential has a spatial range of 2R g and a strength determined by the ideal gas law for the polymer osmotic pressure. Although useful for many situations, the AO model has obvious limi- tations, e.g., it cannot address semidilute solutions, where the macromolecule concentration exceeds the threshold c  p for polymer-polymer interpenetration or variable polymer- solvent interactions (“solvent quality”). Verma and co- workers [5] have recently examined dilute suspensions of colloidal particles dissolved in DNA R  2.5R g  so- lutions in a nearly “athermal” regime. Although fairly successful in dilute DNA solutions, under semidilute con- ditions both the amplitude and spatial range of the de- pletion potential were qualitatively changed due to the reduction of the polymer mesh size or density-density cor- relation length, j r [3]. A much less well-studied regime is motivated by the use of nonadsorbing uncharged polymers to aid protein sepa- rations or crystallization [6]. Here R g R $ 1, resulting in “long range” attractive depletion forces between globular proteins. For protein crystallization, a commonly used nonionic, water soluble polymer is poly(ethylene glycol) (PEG) with the chemical formula — CH 2 -CH 2 -O N 2 . PEG has the useful advantage of interacting weakly with protein molecules [6]. PEGwater solutions phase separate upon heating, giving rise to a lower critical solution temperature (LCST). As the phase boundary is approached, the mesh size grows significantly thus altering the depletion potential between proteins which can be thermodynamically quantified through the second virial coefficient, B 2 . In this Letter we present the first systematic measurements of B 2 in aqueous protein-PEG solutions as a function of three dimensionless variables: size ratio R g R, polymer concentration c p c  p , and reduced temperature T T c , where T c is the critical tem- perature for polymer-solvent phase separation. The globular protein studied is hen egg white lysozyme R  1.7 nm purchased from Seikagaku America Inc., which is used without further purification. The protein was dissolved in salt-water-PEG solutions (sodium acetate buffer at pH 4.6 with ionic strength controlled by the ad- dition of NaCl). All experiments were performed at 25 ± C unless stated otherwise. The lysozyme second virial coef- ficient is directly related to the protein-protein potential of mean force V r  as B 2  2p Z ` 0 r 2 1 2 e 2V r kT  dr . (2) Details of sample preparation and static light scattering methods of measuring B 2 are provided elsewhere [7]. PEG of molecular weights M w  1000, 6000, and 12 000 were purchased from the Sigma Chemical Company. For quan- titative characterization purposes, the osmotic pressure of each PEG solution was measured in the absence of protein but under identical solvent conditions, for polymer con- centrations up to slightly above the semidilute threshold 4554 0031-9007 99  83(22)  4554(4)$15.00 © 1999 The American Physical Society