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Numerical solutions of the Poisson-Boltzmann equation (PBE) have found wide application in the computation of electrostatic energies of hydrated molecules, including biological macromolecules. However, solving the PBE for electrostatic forces has proved more difficult, largely because of the challenge of computing the pressures exerted by a high dielectric aqueous solvent on the solute surface. This paper describes an accurate method for computing these forces. We begin by presenting a novel derivation of the forces acting in a system governed by the PBE. The resulting expression contains three distinct terms: the effect of electric fields on 'fixed" atomic charges; the dielectric boundary pressure, which accounts for the tendency of the high dielectric solvent to displace the low dielectric solute wherever an electric field exists; and the ionic boundary pressure, which accounts for the tendency of the dissolved electrolyte to move into regions of nonzero electrostatic potential. Techniques for extracting each of these three force contributions from finite difference solutions of the PBE for a solvated molecule are then described. Tests of the methods against both analytic and numeric results demonstrate their accuracy. Finally, the electrostatic forces acting on the two members of a salt bridge in the enzyme triose phosphate isomerase are analyzed. The dielectric boundary pressures are found to make substantial contributions to the atomic forces. In fact, their neglect leads to the unphysical situation of a significant net electrostatic force on the system. In contrast, the ionic boundary forces are usually extremely weak at physiologic ionic strength.

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M o r t o n D a v i s T h e C i t y U n i v e r s i t y of N e w Y o r k and M i c h a e l M a s c h l e r T h e H e b r e w U n i v e r s i t y J e r u s a l e m , I s r a e l ABST RAG T The kernel of a cooperative n -p e r s o n game i s defined. It i s a subs e t of the bargaining s e t m:i). Its existence and some of i t s p r o p e r t i e sa r e studied. We apply it to the 3-person games, to the 4-person constants u m games, to the s y m m e t r i c and n-quota g a m e s and to games in which only the n and the (n-1)-person coalitions a r e allowed to be non-flat.In o r d e r to illustrate i t s m e r i t s and d e m e r i t s a s a predictor of an actual outcome in a real-life situation, we exhibit an example in which the k e r n e l prediction s e e m s frustrating. The opinions of other authors a r e quoted in o r d e r to throw some light on this interesting example.authors were staying at the Econometric R e s e a r c h P r o g r a m , Princeton University, Princeton, New J e r s e y .tA coalition is called flat if its value is equal to the sum of the values of its m e m b e r s -considered a s 1-person coalitions. 223x. 2 0, i = 1,2,. . . , n (individual rationality)The symbol (x; 8) will be called an individually rational payoff configuration (i.r.p.c. o r P.c.).If we fix the coalition structure 8, then the set of all the payoffs X, satisfying (2.5) and (2.6), is a Cartesian product of m simplices:(2.7) 1 2 r 1 j = 1 , 2 ,..., m . DEFINITION 2.1: Let (x;%) be an i.r.p.c. for a game r, and let D be an arbitrary coalition. The excess of D with respect to (x;$) is (2.9) e(D) = v(D)xi. ie D The excess of D therefore represents the total amount that the members of D gain (or lose, i f e(D) < O), i f they withdraw from (x;B) and form the coalition D. Clearly, (2.10) e(Bj) = 0, j = 1,2,. . . , m .Let k and P be two distinct players in a coalition B. of 8; we denote by 3 the set J k, P of all the coalitions which contain player k but do not contain player 1 ; i.e., (2.11) 5 {DID CN, keD,P(D}. DEFINITION 2.2: Let (x;%) be an i.r.p.c. for a game r, and let k and P be two distinct players in a coalition B. of 53. The maximum surplus of k over 1 with respect to (x;8) is J (2.12) % Max e(D). DE ' k,P 9The maximum surplus, therefore, represents the maximal amount player k can gain (or the minimal amount that he must lose), by withdrawing from (x;%) and joining a coalition D which does not require the consent of P (since PbD), with the understanding that the other members of D will be satisfied with getting the same amount they had in (x;53). DEFINITION 2.3: JLet (x; 8) be an i.r.p.c. for a game r, and let k,P be two distinct players in a coalition B. of $3. Player k is said to outweigh player P with respect to (x; 53), 226 M. DAVIS AND M. MASCHLER and this is denoted by k >> Q, or, equivalently, by Q << k, if (2.13)If neither k >> P nor Q >> k, we say that k and Q are in equilibrium. For the sake of completeness we define each player to be in equilibrium with himself. Similarly, we also regard any two players, who...