VOL. 14,1685-1700 (1975)
BIOPOLYMERS
Sedimentation of Generalized Systems of Interacting Particles. I. Solution of Systems of Complete Lamm Equations JEAN-MICHEL CLAVERIE, HENRI DREUX,* and RENfi COHEN, Institut de Biologie MolBculaire, Centre National de la Recherche Scientifique, UniversitB Paris VII, 75221 Paris Cedex 05
Synopsis A very general approach to the chemical equilibria between many interacting molecules during sedimentation (boundary, band, or active enzyme) taking into account boundary conditions, cell geometry, equilibrium constants, diffusion, enzyme kinetics, etc., is presented. Through a Fortran program, the method has been applied to two very simple but typical cases. With only minor adjustments, the method presented here for sedimentation studies can be extended to all sorts of problems in which “pools” of various species are interacting with each other.
INTRODUCTION For many years (up to about 1960), the analytical centrifuge was a major tool in biochemistry. It was then mostly used in connection with purity and molecular-weight problems, and required one to work with concentrations of the order of 1 mg/ml (Schlieren and interference optics). For these conceentrations, the theory of centifugation was not an easy one to handle even in the case of a single component. It was even more difficult when equilibria, even as simple as monomer + dimer, were c0ncerned.l The glorious years of molecular biology (after 1953) gave a new impetus to absorption optics (mostly uv), in connection with the CsCl density-gradient type of experiments in the nucleic acid field. This led eventually to the development of commercial uv scanners (with or without the recent use of a computer hooked up to them); their sensitivity is close to 0.01 absorbance unit, permitting one to lower the required protein concentrations. But a t the same time, vast improvements in new techniques (Sephadex, polyacrylamide-sodium-dodecyl-sulfategel), which were more handy than analytical centrifugation, made this last technique almost obsolete. Its main uses (apart from routine work on sera, etc.) in scientific research are in the nucleic acid field (Cs salt technique), sometimes on the molecular weight of native protein, or in very special applications restricted to a few laboratories (conformational changes, etc.). Even * Deceased June 2,1974. 1685 01975 by John Wilev 81 Sons, Inc.
1686
CLAVERIE, DREUX, AND COHEN
the equilibrium technique in urea or guanidine hypochloride has been almost given up for the'lack of a precise theory. So at the end of the sixties, analytical centrifugation was an almost dying technique. As for the theoretical aspect even the simple monomer * dimer system sedimentation has not yet been solved completely, although a good approximate solution has been recently published2 (the cell geometry as well as the field variation with the radius were not taken into account). An entirely novel approach was opened 10 years ago by the active enzyme analytical centrifugation method, which amounts to an enzyme band sedimenting through a substrate solution; the enzymatic reaction being followed, during the centrifugation, through the absorption optics, by using the difference, when it exists, between the absorption spectra of the substrate and the product of the r e a ~ t i o n . The ~ required concentration range was then lowered by at least two orders of magnitude (often to less than 1 fig/ml), and it became possible to study very impure preparations, often a cellular crude extract as it comes, for example, from the sonicator. This technique, though very useful, required very sensitive absorption optics and has been used to its full capacity only since 1971, when a new type of entirely automated scanner, capable of a 0.0001 absorbance unit sensitivity was put into routine use.4 It became immediately apparent that in crude extracts, enzymes had a behavior entirely different from the one they show when purified. We found circumstances where, obviously, a certain enzymatic activity was not carried by a unique molecular-weight structure, but seemed to be exchanged continuously between a very light structure (probably the free enzyme) and a much more massive structure (Claverie, Schmitt, and Cohen, unpublished results). I t was very important to be able to handle the unusual data we were collecting and then, perhaps, to open a new way in the study of the cell organization. At this point, we were confronted with a very important practical choice: either, from these data, to discover the number of components involved in the chemical equilibrium, the sedimentation and diffusion coefficients of each of them, as well as the kinetic constants and relaxation times of these equilibria; or to assume a certain type of equilibrium (model), simulate the data one should get, and compare them with the real data. In theory, the two approaches are equivalent but from the computer point of view, we adopted the second one, and in this series of papers whose starting point was a particular centrifugation method (active enzyme), we present a very general approach to the chemical equilibria between many reacting molecules during sedimentation (boundary, band, or active enzyme) taking into account: 1) the boundary conditions, the true cell geometry, the variation of the centrifugal field with the radius, and the diffusion phenomenon in the case of a single sedimenting species (this paper); 2) the various kinetic constants of the interactions between the system components, including, for the active enzyme centrifugation, the enzymatic kinetics constants (accompanying paper);7 3) '
SEDIMENTATION IN SYSTEMS OF PARTICLES. I
1687
the concentration dependency of s and D (third paper of this series). Our method rests on the use of the “finite element”5 method of solving numerically (by computer) partial differential equations, and can, with only minor adjustments, be extended to all sorts of problems in which “pools” of various species are interacting with each other. Already, in biochemistry, this numerical method has been used for the study of the behavior of the immobilized enzymes.6 Since we are presenting a very general and practical approach to many problems, far beyond the scope of centrifugation theory alone, since this paper is the first of a series that will deal with various and complicated interacting systems, and furthermore since we think that this paper will permit anyone having a problem relevant to the method presented here to solve it easily, it seemed arrogant not to give a t least some glimpses of the manner the problem was mathematically handled, and Lransformed, until it reached a form convenient for a Fortran treatment. As simple examples and as a test of its validity and of its precision, the method presented here has been applied to two typical cases. We tried to present our results in such a way that they can be used easily. The basic program itself (about 400 lines long) is actually a group of subroutines; it has been written in Fortran IV and its length is equal to about (400 20.n) lines, where n is the number of components of interest in the system ( n = 1 for the reference case above, n = 4 for the fourth problem treated in the following paper,7 etc.); 400 is the length of a group of subroutines written once and for all and common to all the problems; 20.n is the number of lines needed to assemble these subroutines. As for the time required by the computer to simulate a 1-hr sedimentation, it was equal to 75.n sec with the medium-size computer we used; it could be reduced by an order of magnitude with other computers. These practical programming considerations, added to what is shown in the present and the following papers, clearly demonstrate the great potentialities and flexibility of the numerical method presented in this paper. One can predict that, in the sedimentation field a t least, very many theoretical problems that are still unsolved, or only partially or approximately solved, should soon receive correct and very instrumental solutions.
+
SETTING THE PROBLEM What is needed is a method able, through the use of a digital computer, to describe the behavior of complex chemical systems during sedimentation. By “complex,” we mean that many species are interacting in various ways, but from the thermodynamical point of view only “ideal” solute(s)-solvent solutions will be considered here. We can define as the reference system, the simplest ideal solution: one solute-one solvent; ; its sedimentation is mathematically described
CLAVERIE, DREUX, AND COHEN
1688
by the classical Lamm equation:’
where
J = sdr-C
C
=
ac
- D-
=
dr
flux
solute concentration
w = angular velocity
D
=
s
=
diffusion coefficient sedimentation coefficient r = radial distance
Starting from this reference system, the “complexity” of any other system studied will be represented by 1) the number of equations required to describe the behavior of the system; it is equal to the number of chemical components (solutes); and 2) the appearance of a nonzero right-hand term in each of these equations, describing the ways these components are coupled. The monomer-polymer system will be given as an example of the way any problem can be set. The sedimentation of such a system, in which the chemical reaction is represented by the scheme:
M
+ M + ... + M
\
”
I
k,
x k-1
n where
M A
= monomer =
polymer (n-mer)
will be described by a system of two equations:
X M+
-
at
ld(rJM> = f r dr
-____
(monomer)
where f = k-lC, - klC&. The kinetic constants kl and k-1 can be functions of r (pressure effects) or not. Such a mathematical formulation shows clearly the superposition of a chemical reaction (right-hand term) to the sedimentation-diffusion process of two distinct species (left-hand term). This system, as well as three-component ones of the type monomer * dimer * tetramer will be considered in a forthcoming paper of the series.
SEDIMENTATION IN SYSTEMS OF PARTICLES. I
1689
This example shows a structure, common to all complex systems: 1) the number of equations isequal to the number of components; 2) each left-hand term is formally identical with the left-hand term of the classical Lamm equations; 3) the right-hand terms can be extremely variable, being usually nonlinear coupling terms. The method thus found will be described; it led to a computer representation of what can be called the “sedimentation-diffusion operator,” which will be shown to be compatible with any type of right-hand term. Thus what is presented here is a numerical solution of systems of complete Lamm equations (i.e., with nonzero right-hand terms), which permits one to get the radial distributions of all the components (simulations).
THE LAMM EQUATION To find a representation of the sedimentation-diffusion operator, one is led to solve the simplest problem, the reference system (above) where the right-hand term of the unique equation is zero. This very specific problem has already been s o l ~ e dbut ;~~ in~view of the fact that we aim a t the most general method able to deal with all sorts of yet unsolved problems, an original numerical method had to be devised.
Mathematical Model and Regularity T o have a completely defined mathematical problem, one has to add certain “boundary conditions” to the Lamm equation. The spatial boundary conditions express the fact that the flux J is always zero a t the meniscus ( r = r,) as well as a t the centrifugation cell bottom ( r = r b ) ; this gives:
J(r,t)
=
J ( r h , t ) = 0;
0
< t < T.
(3)
The time boundary condition states that a t the beginning ( t = 0), the concentration function is imposed by the experimental physical data; this can be expressed by:
where & ( r ) is a “sufficiently regular” function of the spatial coordinates (for example, such that n
J
Ct(r)rdr
BIOPOLYMERS
Sedimentation of Generalized Systems of Interacting Particles. I. Solution of Systems of Complete Lamm Equations JEAN-MICHEL CLAVERIE, HENRI DREUX,* and RENfi COHEN, Institut de Biologie MolBculaire, Centre National de la Recherche Scientifique, UniversitB Paris VII, 75221 Paris Cedex 05
Synopsis A very general approach to the chemical equilibria between many interacting molecules during sedimentation (boundary, band, or active enzyme) taking into account boundary conditions, cell geometry, equilibrium constants, diffusion, enzyme kinetics, etc., is presented. Through a Fortran program, the method has been applied to two very simple but typical cases. With only minor adjustments, the method presented here for sedimentation studies can be extended to all sorts of problems in which “pools” of various species are interacting with each other.
INTRODUCTION For many years (up to about 1960), the analytical centrifuge was a major tool in biochemistry. It was then mostly used in connection with purity and molecular-weight problems, and required one to work with concentrations of the order of 1 mg/ml (Schlieren and interference optics). For these conceentrations, the theory of centifugation was not an easy one to handle even in the case of a single component. It was even more difficult when equilibria, even as simple as monomer + dimer, were c0ncerned.l The glorious years of molecular biology (after 1953) gave a new impetus to absorption optics (mostly uv), in connection with the CsCl density-gradient type of experiments in the nucleic acid field. This led eventually to the development of commercial uv scanners (with or without the recent use of a computer hooked up to them); their sensitivity is close to 0.01 absorbance unit, permitting one to lower the required protein concentrations. But a t the same time, vast improvements in new techniques (Sephadex, polyacrylamide-sodium-dodecyl-sulfategel), which were more handy than analytical centrifugation, made this last technique almost obsolete. Its main uses (apart from routine work on sera, etc.) in scientific research are in the nucleic acid field (Cs salt technique), sometimes on the molecular weight of native protein, or in very special applications restricted to a few laboratories (conformational changes, etc.). Even * Deceased June 2,1974. 1685 01975 by John Wilev 81 Sons, Inc.
1686
CLAVERIE, DREUX, AND COHEN
the equilibrium technique in urea or guanidine hypochloride has been almost given up for the'lack of a precise theory. So at the end of the sixties, analytical centrifugation was an almost dying technique. As for the theoretical aspect even the simple monomer * dimer system sedimentation has not yet been solved completely, although a good approximate solution has been recently published2 (the cell geometry as well as the field variation with the radius were not taken into account). An entirely novel approach was opened 10 years ago by the active enzyme analytical centrifugation method, which amounts to an enzyme band sedimenting through a substrate solution; the enzymatic reaction being followed, during the centrifugation, through the absorption optics, by using the difference, when it exists, between the absorption spectra of the substrate and the product of the r e a ~ t i o n . The ~ required concentration range was then lowered by at least two orders of magnitude (often to less than 1 fig/ml), and it became possible to study very impure preparations, often a cellular crude extract as it comes, for example, from the sonicator. This technique, though very useful, required very sensitive absorption optics and has been used to its full capacity only since 1971, when a new type of entirely automated scanner, capable of a 0.0001 absorbance unit sensitivity was put into routine use.4 It became immediately apparent that in crude extracts, enzymes had a behavior entirely different from the one they show when purified. We found circumstances where, obviously, a certain enzymatic activity was not carried by a unique molecular-weight structure, but seemed to be exchanged continuously between a very light structure (probably the free enzyme) and a much more massive structure (Claverie, Schmitt, and Cohen, unpublished results). I t was very important to be able to handle the unusual data we were collecting and then, perhaps, to open a new way in the study of the cell organization. At this point, we were confronted with a very important practical choice: either, from these data, to discover the number of components involved in the chemical equilibrium, the sedimentation and diffusion coefficients of each of them, as well as the kinetic constants and relaxation times of these equilibria; or to assume a certain type of equilibrium (model), simulate the data one should get, and compare them with the real data. In theory, the two approaches are equivalent but from the computer point of view, we adopted the second one, and in this series of papers whose starting point was a particular centrifugation method (active enzyme), we present a very general approach to the chemical equilibria between many reacting molecules during sedimentation (boundary, band, or active enzyme) taking into account: 1) the boundary conditions, the true cell geometry, the variation of the centrifugal field with the radius, and the diffusion phenomenon in the case of a single sedimenting species (this paper); 2) the various kinetic constants of the interactions between the system components, including, for the active enzyme centrifugation, the enzymatic kinetics constants (accompanying paper);7 3) '
SEDIMENTATION IN SYSTEMS OF PARTICLES. I
1687
the concentration dependency of s and D (third paper of this series). Our method rests on the use of the “finite element”5 method of solving numerically (by computer) partial differential equations, and can, with only minor adjustments, be extended to all sorts of problems in which “pools” of various species are interacting with each other. Already, in biochemistry, this numerical method has been used for the study of the behavior of the immobilized enzymes.6 Since we are presenting a very general and practical approach to many problems, far beyond the scope of centrifugation theory alone, since this paper is the first of a series that will deal with various and complicated interacting systems, and furthermore since we think that this paper will permit anyone having a problem relevant to the method presented here to solve it easily, it seemed arrogant not to give a t least some glimpses of the manner the problem was mathematically handled, and Lransformed, until it reached a form convenient for a Fortran treatment. As simple examples and as a test of its validity and of its precision, the method presented here has been applied to two typical cases. We tried to present our results in such a way that they can be used easily. The basic program itself (about 400 lines long) is actually a group of subroutines; it has been written in Fortran IV and its length is equal to about (400 20.n) lines, where n is the number of components of interest in the system ( n = 1 for the reference case above, n = 4 for the fourth problem treated in the following paper,7 etc.); 400 is the length of a group of subroutines written once and for all and common to all the problems; 20.n is the number of lines needed to assemble these subroutines. As for the time required by the computer to simulate a 1-hr sedimentation, it was equal to 75.n sec with the medium-size computer we used; it could be reduced by an order of magnitude with other computers. These practical programming considerations, added to what is shown in the present and the following papers, clearly demonstrate the great potentialities and flexibility of the numerical method presented in this paper. One can predict that, in the sedimentation field a t least, very many theoretical problems that are still unsolved, or only partially or approximately solved, should soon receive correct and very instrumental solutions.
+
SETTING THE PROBLEM What is needed is a method able, through the use of a digital computer, to describe the behavior of complex chemical systems during sedimentation. By “complex,” we mean that many species are interacting in various ways, but from the thermodynamical point of view only “ideal” solute(s)-solvent solutions will be considered here. We can define as the reference system, the simplest ideal solution: one solute-one solvent; ; its sedimentation is mathematically described
CLAVERIE, DREUX, AND COHEN
1688
by the classical Lamm equation:’
where
J = sdr-C
C
=
ac
- D-
=
dr
flux
solute concentration
w = angular velocity
D
=
s
=
diffusion coefficient sedimentation coefficient r = radial distance
Starting from this reference system, the “complexity” of any other system studied will be represented by 1) the number of equations required to describe the behavior of the system; it is equal to the number of chemical components (solutes); and 2) the appearance of a nonzero right-hand term in each of these equations, describing the ways these components are coupled. The monomer-polymer system will be given as an example of the way any problem can be set. The sedimentation of such a system, in which the chemical reaction is represented by the scheme:
M
+ M + ... + M
\
”
I
k,
x k-1
n where
M A
= monomer =
polymer (n-mer)
will be described by a system of two equations:
X M+
-
at
ld(rJM> = f r dr
-____
(monomer)
where f = k-lC, - klC&. The kinetic constants kl and k-1 can be functions of r (pressure effects) or not. Such a mathematical formulation shows clearly the superposition of a chemical reaction (right-hand term) to the sedimentation-diffusion process of two distinct species (left-hand term). This system, as well as three-component ones of the type monomer * dimer * tetramer will be considered in a forthcoming paper of the series.
SEDIMENTATION IN SYSTEMS OF PARTICLES. I
1689
This example shows a structure, common to all complex systems: 1) the number of equations isequal to the number of components; 2) each left-hand term is formally identical with the left-hand term of the classical Lamm equations; 3) the right-hand terms can be extremely variable, being usually nonlinear coupling terms. The method thus found will be described; it led to a computer representation of what can be called the “sedimentation-diffusion operator,” which will be shown to be compatible with any type of right-hand term. Thus what is presented here is a numerical solution of systems of complete Lamm equations (i.e., with nonzero right-hand terms), which permits one to get the radial distributions of all the components (simulations).
THE LAMM EQUATION To find a representation of the sedimentation-diffusion operator, one is led to solve the simplest problem, the reference system (above) where the right-hand term of the unique equation is zero. This very specific problem has already been s o l ~ e dbut ;~~ in~view of the fact that we aim a t the most general method able to deal with all sorts of yet unsolved problems, an original numerical method had to be devised.
Mathematical Model and Regularity T o have a completely defined mathematical problem, one has to add certain “boundary conditions” to the Lamm equation. The spatial boundary conditions express the fact that the flux J is always zero a t the meniscus ( r = r,) as well as a t the centrifugation cell bottom ( r = r b ) ; this gives:
J(r,t)
=
J ( r h , t ) = 0;
0
< t < T.
(3)
The time boundary condition states that a t the beginning ( t = 0), the concentration function is imposed by the experimental physical data; this can be expressed by:
where & ( r ) is a “sufficiently regular” function of the spatial coordinates (for example, such that n
J
Ct(r)rdr
