Abstract:
A method de novo is provided for predicting crystallization conditions and for crystallizing biomacromolecules, in particular proteins. The method provides a simple, quick and precise approach in determining the biomacromolecule solubility in different solutions, as well as the boundary between crystallization and aggregation. Because the method relies only on monitoring the assembly behavior of the biomacromolecule at the surface of a solution, it has general applicability and requires a relatively short amount of time to provide results that are reliable. Because there is no need to first crystallize the biomacromolecule, small amounts of protein are sufficient. Because the method works by measuring the surface tension or pressure of the surface of the biomacromolecule solution, it is easy, precise and quick. Furthermore it is cost-effective in requiring simple and inexpensive equipment.

Description:
FIELD OF INVENTION  
       [0001]     This invention relates to a de novo method of predicting crystallization conditions. In particular, the invention relates to a method of predicting the conditions for crystallizing biomacromolecules from solution, and more specifically for crystallizing proteins from solution.  
       BACKGROUND  
       [0002]     In most cases of protein crystallization experiments, the final product is not a single crystal but amorphous aggregation. To predict the likelihood of the formation of either a crystal or an amorphous aggregation, the second virial coefficient B 22  is customarily employed by many groups [1-5]. Serving as an indicator of intermolecular interactions, B 22  is positive when these interactions are repulsive, and negative when these interactions are attractive. As a consequence, a necessary condition for crystallization is obtained when the second virial coefficient B 22  lies in a so-called “crystallization window”: −8×10 −4 &lt;B 22 &lt;−2×10 −4  ml mol/g 2 .  
         [0003]     The criterion based on the second virial coefficient has its advantages, as it gives a discriminating response. However this criterion does not work in some cases in which B 22  could well lie within the crystallization window, but the experiment gives amorphous aggregation. One of the reasons for this failure is that the criterion provided by the second virial coefficient takes into account only the interactions between the biomacromolecules. However, such intermolecular interactions determine biomacromolecule crystallization only partially. Apart from this, the B 22  criterion only attempts to determine the solvent conditions for which crystallization rather than amorphous aggregation would take place. However, for biomacromolecule crystallization to be possible in the first place, an additional condition must be satisfied, namely that the biomacromolecule concentration must exceed its equilibrium value in the context of the applied solvent conditions. The B 22  criterion does not address this problem, and gives no information whatsoever on the critical equilibrium condition of the biomacromolecule.  
         [0004]     The crystallization of biomacromolecules involves a nucleation and growth process, determined to a large extent by kinetics. Kinetics refers to the way biomacromolecules move in a solution, the rate at which they are transported, and the way they are incorporated in the biomacromolecule crystals at the crystal surface. The crystallization window provided by the second virial coefficient disregards kinetic and other factors, that are unrelated to intermolecular interactions but nevertheless largely influence crystallization.  
         [0005]     Alternative methods have been developed to circumvent the drawbacks presented by the use of the second virial coefficient in attempting to predict crystallization. These methods, the so-called high throughput screening methods, are solely empirical involving a large number of solution matrices. These methods are costly, cumbersome, and time-consuming; they require large investments in expensive robots. As protein crystallization is normally a lengthy process and can be affected by about 20 different parameters, these methods have in many cases no general applicability and suffer from a low success rate.  
         [0006]     What is needed is a de novo method offering a twofold advantage above the prior art. Firstly, such a method should determine the equilibrium biomacromolecule concentration, so that the values of the biomacromolecule concentration used in the process of crystallization in the prevailing experimental conditions can be restricted to values above the equilibrium value. Secondly, such a method should predict more reliably the crystallization conditions referring to the solvent experimental parameters. Finally, such a method should have a more general applicability and be simple, easy to apply, not wasteful on the biomacromolecule or the protein. It should not require complex equipment, or rely on heavy investments for its application, and it should be amenable to being utilized by both the institutional and industrial establishments, while at the same time offering substantial automation advantages, and being much less costly than the methods recited in the prior art. Such a de novo method does not exist in the prior art.  
       REFERENCES  
       [0000]    
       
          R. A. Curtis, J. M. Prausnitz et al., Protein-protein and protein-salt interactions in aqueous protein solutions containing concentrated electrolytes, Biotechnol. Bioeng, 57: 11-21(1998).  
          Y. C. Chiew, D. Kuehner et al., Molecular thermodynamics for salt-induced protein precipitation, AIChE J. 41: 2150-2159 (1995).  
          A. George and W. W. Wilson, Predicting protein crystallization from a dilute solution property, Acta Crystallogr. D. 50: 361-365 (1994).  
          B. L. Neal and D. Asthagiri et al., Why is the osmotic second virial coefficient related to protein crystallization? J. Crystal Growth, 196: 377-387 (1999).  
          F. Bonnete and S. Finet et al., Second virial coefficient variations with lysozyme crystallization conditions, J. Crystal Growth, 196: 403-414 (1999).  
          E. Tornberg, The application of the drop volume technique to measurement of the adsorption of protein at interfaces, J. Colloid Interface Sci. 64(3): 391-402 (1978).  
          D. E. Graham and M. C. Phillips, Protein at liquid interfaces I: kinetics of adsorption and surface denaturation, J. Colloid Interface Sci. 70(3): 403-417 (1979).  
          M. Subirade et al., Effect of dissociated and conformational changes on the surface behavior of pea legumin, J. Colloid Interface Sci. 152(2): 442-454 (1992).  
       
     
       SUMMARY  
       [0015]     A de novo method for predicting crystallization conditions for biomacromolecules is provided, giving a more reliable prediction criterion than has been possible thus far. Unlike the second virial coefficient B 22  used to predict crystallization conditions based solely on intermolecular interactions, the present invention combines information on both the intermolecular interactions and kinetic effects to prescribe crystallization conditions. The improvement above the empirical methods available in the prior art and the prediction methods provided by calculating the second virial coefficient, is due to the incorporation of the kinetic effects that largely determine biomacromolecule crystallization, and offers the following advantages: (1) it establishes an equilibrium macromolecule concentration that should be exceeded by the applied macromolecule concentration used for crystallization; (2) only small amounts of protein are required; (3) the measurements are much quicker than either through use of the B 22  approach or through use of the empirical screening methods; (4) the simplicity of the required measurement of the assembly parameter (e.g. surface tension) of the biomacromolecule solution facilitates the handling in an ordinary laboratory.  
         [0016]     One object of the present invention is to identify the phase boundary between a crystal phase and a liquid phase by determining the biomacromolecule solubility, under prevailing experimental conditions. Another object of the invention is to identify the boundary between a crystallization and an aggregation under prevailing experimental conditions.  
         [0017]     Therefore in accordance with a first aspect of the invention there is disclosed:  
         [0000]     A method for predicting a crystal equilibrium condition for biomacromolecule crystallization and for crystallizing a biomacromolecule, comprising  
         [0000]     setting up at least one biomacromolecule solubility experiment comprising the steps of  
         [0000]    
       
         
           
              a) preparing a solution of the biomacromolecule in a solvent, the solution having a biomacromolecule concentration,  
              b) selecting a variable quantity,  
              c) selecting an assembly parameter,  
              d) monitoring a response of the assembly parameter while varying the variable quantity in a suitable way so that the response exhibits a transition,  
              e) obtaining an equilibrium biomacromolecule concentration based on the transition,  
              f) defining a crystal equilibrium condition according to which a biomacromolecule crystallization concentration exceeds the equilibrium biomacromolecule concentration, 
 
 and crystallizing the biomacromolecule. 
 
           
         
       
     
         [0024]     In accordance with a second aspect of the invention there is disclosed:  
         [0000]     A method for predicting an aggregation boundary condition for biomacromolecule crystallization and for crystallizing a biomacromolecule, comprising  
         [0000]     setting up at least one aggregation boundary condition experiment comprising  
         [0000]    
       
         
           
              a) preparing a solution of the biomacromolecule,  
              b) selecting a variable quantity,  
              c) selecting an assembly parameter,  
              d) measuring the assembly parameter at different times,  
              e) registering an equilibrium assembly parameter  
              f) deriving a crystallization coefficient from the equilibrium assembly parameter, the crystallization coefficient being associated with the variable quantity,  
              g) using an aggregation indicator to define an aggregation boundary condition for the biomacromolecule, the aggregation boundary condition prescribing that an aggregation occurs when the crystallization coefficient associated with the variable quantity is larger than the aggregation indicator, 
 
 and crystallizing the biomacromolecule.
 
           
         
       
     
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0032]     Embodiments of the present invention are illustrated by the following drawings:  
         [0033]      FIG. 1  is an illustration of predictive conditions;  
         [0034]      FIG. 1A  is an illustration of a crystal equilibrium condition, leading to an equilibrium biomacromolecule concentration;  
         [0035]      FIG. 1B  is an illustration of an aggregation boundary condition, leading to a crystallization coefficient;  
         [0036]      FIG. 2  is an illustration of a crystallization process from a solution of biomacromolecules, involving a transition from a liquid phase to an orderly solid phase, that is a crystal;  
         [0037]      FIG. 3  is an illustration of an amorphous aggregation process involving the transition from the liquid phase to a disorderly solid phase;  
         [0038]      FIG. 4  is an illustration of analogous processes to the crystallization and the amorphous aggregation that take place in the solution;  
         [0039]      FIG. 5  is an illustration of a response of an assembly parameter to a tendency of the biomacromolecules to assembly;  
         [0040]      FIG. 6  is an illustration of the way a solubility curve relates to the crystal equilibrium condition.  
         [0041]      FIG. 7  shows plots of the assembly parameter against time, when the solution of biomacromolecules includes a protein lysozyme and a salt sodium chloride;  
         [0042]      FIG. 8  shows plots of the assembly parameter against the square root of the time;  
         [0043]      FIG. 9  shows plots of the logarithm of the time derivative of the assembly parameter;  
         [0044]      FIG. 10  is an illustration of a time-dependence profile of a macromolecule behavior;  
         [0045]      FIG. 11  shows plots of a diffusion time, a penetration time and a rearrangement time;  
         [0046]      FIG. 12  shows a histogram illustrating how crystallization coefficients relate to the aggregation boundary condition by prescribing an aggregation indicator;  
         [0047]      FIG. 13  shows a plot of the assembly parameter against time, when the solution of biomacromolecules includes the protein lysozyme and a salt ammonium sulfate;  
         [0048]      FIG. 14  shows a plot of the assembly parameter against time, when the solution of biomacromolecules includes a protein concalavine A and the salt ammonium sulfate; and  
         [0049]      FIG. 15  shows plots of the assembly parameter against time, when the solution of biomacromolecules includes a protein bovine serum albumin (BSA) and the salt ammonium sulfate. 
     
    
     DETAILED DESCRIPTION  
       [0050]     The primary features of a first embodiment of the invention of a method of predicting biomacromolecule crystallization conditions and for crystallizing biomacromolecules are provided hereinafter with reference to  FIG. 1A  in  FIGS. 2-6 . The primary features of a second embodiment of the invention are provided hereinafter with reference to  FIG. 1B  in  FIGS. 2-5  and  FIGS. 7-15 . In the first embodiment, a crystal equilibrium condition in  FIG. 1A  is expressed by means of a macromolecule solubility curve serving as a boundary separating two regions of experimental parameter values: a region where crystallization can occur, and a region where crystallization cannot occur.  
         [0051]     The method described in the first embodiment establishes a biomacromolecule equilibrium concentration in the context of the applied experimental conditions. The biomacromolecule concentration to be used for crystallization must exceed the obtained equilibrium value. In a second embodiment, an aggregation boundary condition in  FIG. 1B  is expressed by means of a window of experimental parameter values above which the amorphous aggregation is likely to occur. The terms “aggregation” and “amorphous aggregation” are used interchangeably.  
         [0000]     Theoretical Introduction  
         [0052]     In attempts to crystallize biomacromolecules from a solution, it is desirable to obtain as much as possible single crystals with as few defects as possible, and to avoid amorphous aggregations of molecules, since amorphous aggregates are not crystals. The present invention takes advantage of the property of biomacromolecules to have mixed hydrophobic and hydrophilic regions. This property results in a tendency for these molecules to assembly either in the bulk or at the surface of the solution. In this disclosure, the surface of the solution can be adjacent to another material or to empty space, and hence the surface can be in contact with a solid or with a liquid or with a gas, that is usually air. The surface of the solution has a surface tension and a surface pressure, which terms in this case include an interfacial tension or an interfacial pressure.  
         [0053]     It is possible to define one or more assembly parameters that reach a critical response as increasingly more molecules participate in assembly formation. For example, the tendency of biomacromolecules to assembly in a solution can be monitored by taking density, conductivity, detergency, osmotic pressure, surface tension or surface pressure measurements of the solution.  
         [0054]     Biomacromolecule crystallization conditions are reflected in the tendency of biomacromolecules to assembly. The crystallization conditions and the assembly characteristics of biomacromolecules are governed by both the intermolecular forces and the kinetic effects [6-8], depending on the experimental situation. The present invention prescribes two procedures of measuring the assembly parameters in solution so as to determine reliable crystallization conditions without the prior need to carry out crystallization experiments. Without limitation to the scope of the invention, the present examples illustrating the invention show that the disclosed method is particularly applicable when the biomacromolecule is not prone to severe unfolding at the surface of the solution or at an air/solution interface.  
         [0000]     Application  
         [0055]     In  FIG. 2 a  crystallization process from the solution  120  comprising biomacromolecules  130  involves a transition  122  from a liquid phase  124  to an orderly solid phase, that is a crystal  126 , wherein solid-phase biomacromolecules  132  are regularly arrayed and regularly oriented. The biomacromolecules  130  in the liquid phase  124 , become adsorbed biomacromolecules  136  on a surface  138  of the crystal  126 , and subsequently they acquire a suitable orientation  140  in order to become incorporated as in the crystal  126  as the solid-phase biomacromolecules  132 . The crystallization occurs because a transport time, being an amount of time required for the biomacromolecules  130  in the solution  120  to reach the surface  138  of the crystal  126 , otherwise known as a diffusion time t diff , is longer than an integration time t int , wherein the integration time is an amount of time required for the adsorbed biomacromolecules  136  to become the solid-phase biomacromolecules  132 , and it includes an amount of time required for their rearrangement.  
         [0056]     In  FIG. 3  on the other hand, the amorphous aggregation involves the transition  122  from the liquid phase  124  to a disorderly solid phase  128 , wherein aggregate biomacromolecules  134  have irregular positions and irregular orientations. The amorphous aggregation occurs because the diffusion time t diff  required for the biomacromolecules  130  in the solution  120  to reach the solid phase  128  is shorter than the time required for their rearrangement and integration.  
         [0057]     In  FIGS. 2 and 3  the solution  120  has an appropriate temperature and an appropriate pH. In the invention there is no restriction on the appropriate temperature, which is usually taken to be around a room temperature. A buffering material may be introduced in the solution  120  to maintain a desirable level of the pH. The solution  120  can also comprise additives like small or large organic molecules or salt. By way of example, the biomacromolecule  130  in  FIGS. 2 and 3  is a protein, more specifically lysozyme (about 14.3 kDalton) or concalavine A (about 23 kdalton), and the additive is a salt, more specifically sodium chloride or ammonium sulfate.  
         [0058]      FIG. 4  illustrates that processes analogous to the crystallization or the amorphous aggregation take place in the solution  120  containing the biomacromolecules  130 . The solution  120  is placed in a container  140 , and the solution has a surface  152 , which in this application is adjacent to air. During a diffusion step  142 , the biomacromolecules  130  take an average amount of time, that is the diffusion time t diff , to diffuse  148  towards the surface  152 . Subsequently, during a penetration step  144 , the biomacromolecules  130  take an average amount of time, that is a penetration time t pen , to penetrate  150  the surface  152 . Subsequently, during a rearrangement step  146 , the biomacromolecules  130 , that penetrated  150  the surface  152  take an average amount of time, that is a rearrangement time t arr , to undergo a configurational rearrangement  154  and finally become integrated in the surface  152 . Thus the integration time t int  is the sum of the penetration time and the rearrangement time, t int =t pen +t arr , because it equals an amount of time required to complete the penetration step  144  and the rearrangement step  146 .  
         [0059]     In  FIG. 5  the biomacromolecules  130  have a biomacromolecule concentration  156 , and the additives have an additive concentration  157 . Without limitation to the scope of the invention, in the experimental runs of the first embodiment of the invention, the biomacromolecule concentration  156  is taken, e.g. 4 mg/ml lysozyme, while the additive (e.g. salt) concentration  157  is allowed to vary. In some other experimental runs of the first embodiment of the invention, the additive concentration  157  is taken, e.g. 1 M (as in  FIG. 5 ) or 2 M salt, while the biomacromolecule concentration  156  is allowed to vary.  
         [0060]     Referring to  FIGS. 5 and 6 , in the first embodiment of the invention, measurements of an assembly parameter  164  (for example a surface tension or a surface pressure) with respect to a variable quantity  158  (for example the biomacromolecule concentration  156 , the additive concentration  157 , the pH or the temperature), are utilized to express the crystal equilibrium condition. According to that condition, as will be explained further on, in  FIG. 1A  ( FIG. 6 ), the crystallization is expected in a region  172  delineated by (and lying above and to the right of) a biomacromolecule solubility curve  170 .  
         [0061]     In the second embodiment of the invention, a probability that the aggregation ( FIG. 3 ) will dominate above the crystallization ( FIG. 2 ), increases when the integration time t int  is substantially longer than the diffusion time t diff . That probability can be quantified as illustrated in  FIG. 1B  ( FIG. 12 ), by means of a crystallization coefficient  232  k cryst =t int /t diff  defined as a ratio of the integration time to the diffusion time. Good quality crystallization occurs when k cryst  is in a suitable range. The higher the value of k cryst , the higher the probability that the amorphous aggregation will occur. When above k cryst , the solution  120  containing the biomacromolecule  130  exists in a so-called “steady state”, in which neither the crystallization ( 126  in  FIG. 2 ) nor the aggregation ( 128  in  FIG. 3 ) can take place. The aggregation indicator resulting from the crystallization coefficient should therefore certainly lie well above k cryst =1; the likely reason for this is that if k cryst &lt;1, the diffusion step  142  of the biomacromolecules  130  in the solution  120  lasts such a long time, that it becomes difficult for the biomacromoles  130  to come in contact and group with one another. In practice however, in our experimental applications, it turns out that a realistic minimum value for k cryst , allowing biomacromoles  130  to come in contact and group lies around 4 or 4.5.  
         [0062]     Because the crystallization coefficient  232  is a dimensionless ratio, it is expected to have general applicability for most biomacromolecules. The method is illustrated in this experiment by choosing as examples the protein lysozyme to serve as a model system for the biomacromolecule crystallization, as well as the protein concanavalin A. It is therefore expected that the aggregation indicator obtained from the crystallization coefficient derived for the proteins lysozyme and concanavalin A will serve as a standard criterion to define the aggregation boundary condition for the crystallization of most biomacromolecules.  
         [0063]     In the second embodiment of the invention, the solution  120  in  FIGS. 4 and 5  is fairly dilute. Without limitation to the scope of the invention, in some experimental runs the biomacromolecule concentration is taken  156 ,  158  in  FIGS. 4 and 5  is taken in the range 0.01 mg/ml to about 1.2 mg/ml. Measurements of the diffusion time t diff , the penetration time t pen  and the rearrangement time t arr  are used to calculate the crystallization coefficient  232  k cryst =t int /t diff  which in turn is used to express the aggregation boundary condition, as explained further below with reference to  FIGS. 7-15 .  
       First Embodiment  
     Crystal Equilibrium Condition  
       [0064]     In the first embodiment of the invention, in  FIGS. 4-6 , the crystal equilibrium condition is determined by constructing the biomacromolecule solubility curve  170 .  
         [0065]      FIG. 5  illustrates a response of the assembly parameter  164  to the activity of the biomacromolecules  130  in the solution  120 . A suitable assembly parameter  164  for this application is the surface tension or the surface pressure, and in this case it is taken to be the surface tension, that can be measured with a tensiometer. The surface pressure p is related to the surface tension s by the equation: p=s−s 0 , where so is the surface tension of the solvent in the absence of biomacromolecules. A suitable variable quantity  158  for this application is the biomacromolecule concentration  156  or the additive concentration  157 , the pH, or the temperature. When a suitable variable quantity  158  has been selected, it can be made to vary in order to sample the response of the assembly parameter  164 . In this case the variable quantity  158  is taken as the biomacromolecule concentration  156 , whereas the additive concentration  157  is taken to be 1 M salt, e.g. NaCl.  
         [0066]      FIG. 5  shows a plot of the assembly parameter  164  (the surface tension) against the logarithm  159  of the variable quantity  158  (the biomacromolecule concentration  156 ), according to which the assembly parameter  164  decreases as the variable quantity  158  is gradually increased. This is illustrated by the diffusion step  142 , followed by the penetration step  144 , subsequently resulting in completion of the rearrangement step  146 . Upon completion of the rearrangement step  146 , a complete layer of rearranged biomacromolecules has been assembled at the surface  152 , allowing no space for any more biomacromolecules  130  to penetrate the surface  152 . Therefore a further increase of the variable quantity  158 , being in this experiment the biomacromolecule concentration, cannot cause a further decrease in the assembly parameter  164 , being in this experiment the surface tension, but will instead result in accumulation of biomacromolecules  130  in the bulk of the solution  120 .  
         [0067]      FIG. 5  illustrates a biomacromolecule solubility experiment. Thus in  FIG. 5  as more and more biomacromolecules  130  tend towards assembly formation, the response of the assembly parameter  164  shows a transition  162  between a changing response  161  and a substantially constant response  163  of the assembly parameter  164 . The transition  162  is associated not only with a critical response of the assembly parameter  164 , but also with a critical magnitude  165  of the variable quantity  158 . In this case the critical response of the assembly parameter  164  is equal to the surface tension of 53 mN/m. The critical magnitude  165  of the variable quantity  158  is equal to the lysozyme concentration of 4 mg/ml; this corresponds to the logarithm  159  of the variable quantity  158  equal to 1.4, when the additive concentration  157  is equal to 1 M NaCl.  
         [0068]     Because in this particular example the assembly parameter  164  is the surface tension, it decreases as the variable quantity  158 , that is the biomacromolecule concentration  156 , is increased, and hence the critical response of the assembly parameter  164  is substantially minimal. The crystal equilibrium condition can be expressed by means of the critical magnitude  165  of the variable quantity  158 : when the temperature and pH are held at their predetermined values, and when the salt concentration is held at 1 M NaCl, then the crystallization cannot occur for values of the lysozyme concentration  156  falling below the equilibrium biomacromolecule concentration, the value of which in the prevailing experimental conditions is 4 mg/ml. A similar critical behavior is observed when the additive concentration  157  is allowed to vary, while the biomacromolecule concentration  156  is held constant at 4 mg/ml lysozyme (drawing not shown).  
         [0069]     The aforementioned behavior should not be construed to be typical of a general case covered by the scope of the invention in which the transition  162  refers to the changing response and the substantially stable or the substantially unchanging response, considering that the changing response need not imply a purely decreasing or a purely increasing response.  
         [0070]      FIG. 6  illustrates how the biomacromolecule solubility curve  170  can be constructed in order to express the crystal equilibrium condition. The solubility curve  170  follows when two of the following quantities: the biomacromolecule concentration  156 , the additive concentration  157 , the pH and the temperature, are allowed to form a pair of the variable quantities. In the example of  FIG. 6  the biomacromolecule concentration  156  and the additive concentration  157  form the pair of the variable quantities, while the pH is kept fixed. A series of transitions  162  is obtained, each transition associated with the critical response of the assembly parameter  164 . In the particular experiment described here, six transitions  162  were measured and employed in order to construct the solubility curve  170 . For example, a point  171  on the solubility curve  170  indicates the following pair of the variable quantities: the biomacromolecule concentration  156  is 4 mg/ml lysozyme while the additive concentration  157  is around 1 M NaCl.  
         [0071]     The corresponding crystal equilibrium condition follows by specifying that the crystallization can occur in a crystallization region  172  in which either one of the pair of the variable quantities assumes values at or above the corresponding critical magnitude  165  on the solubility curve  170 . Therefore the crystal equilibrium condition as illustrated in  FIG. 6  is obtained by recording a series of the critical magnitudes  165  of the variable quantity and the corresponding critical response of the assembly parameter  164 , as illustrated in  FIG. 5 .  
         [0072]     When the biomacromolecule concentration  156  is used as a variable quantity, the critical magnitude  165  is the equilibrium biomacromolecule concentration. When some other parameter is used as a variable quantity, e.g. one of the additive (salt) concentration  156  or the pH, or the temperature, the equilibrium biomacromolecule concentration is, in the context of the employed experimental conditions, given by the applied biomacromolecule concentration  156 . In either case the crystal equilibrium condition prescribes that for crystallization to take place, the crystallization biomacromolecule concentration must exceed the equilibrium biomacromolecule concentration resulting from the biomacromolecule solubility experiment.  
         [0073]     For the measurement of the points on the solubility curve the Wilhelmy plate method is employed using a K14 Kruss tensiometer, according to the following steps.  
         [0074]     1. A buffer solution is prepared at the predetermined pH. A biomacromolecule stock solution is prepared by dissolving the biomacromolecule in the buffer solution. An additive stock solution is prepared by dissolving the additive in the buffer solution.  
         [0075]     2. In  FIG. 5 , various solutions  120  are prepared at the predetermined temperature, usually room temperature, having the additive concentration  157  and different biomacromolecule concentrations  156  starting from 0 and increasing as long as the solutions  120  maintain a clear appearance. The solutions  120  are kept at the predetermined temperature for a few hours. The surface tension or surface pressure  164  of each solution  120  is measured by putting it into the Kruss tensiometer until the surface tension or pressure reaches a constant value. The chamber of the tensiometer is saturated with pure water vapor to maintain a uniform humidity, and the solutions  120  are kept at the predetermined temperature.  
         [0076]     The surface tension or surface pressure  164  is recorded and plotted against the biomacromolecule concentration  156 . The critical magnitude  165  of the biomacromolecule concentration  156  occurring at the critical point  162  is registered. It corresponds to the surface tension or surface pressure  164 , undergoing the transition  162  from the changing response  161  to the substantially constant response  163  as the biomacromolecule concentration  156  increases. The critical point  162  determined in this way is a solubility value of the biomacromolecule at the additive concentration  157 .  
         [0077]     Step 2 is repeated to obtain solubility points at different additive concentrations  157 . Steps 1 and 2 are repeated to find the solubility values at different additive concentration  157  and different pH.  
       Second Embodiment of the Invention  
     Aggregation Boundary Condition  
       [0078]     In the second embodiment of the invention, the measurement of the aggregation boundary condition is carried out, as illustrated in  FIGS. 4 and 5 , by the following steps.  
         [0079]     1. The buffer solution is prepared at the predetermined pH. The biomacromolecule stock solution is prepared by dissolving the biomacromolecule in the buffer solution. The additive stock solution is prepared by dissolving the additive in the buffer solution.  
         [0080]     2. Various solutions  120  are prepared having the biomacromolecule concentration  156  and different additive concentrations  157 . As already stated, the biomacromolecule solution  120  in  FIG. 4 , used in the second embodiment for obtaining the aggregation boundary condition, should be fairly dilute. In the particular experiments described here the protein concentrations  130  are e.g. 0.01, 0.1 or 1.0 mg/ml. Immediately after mixing each of the solutions  120 , the surface tension is measured with respect to time and recorded. The measurement is stopped when the surface tension in each solution reaches a constant value. The obtained data are analyzed according to the steps outlined in  FIGS. 7-15  described below.  
         [0000]     Step 2 is repeated for different pH.  
         [0081]     In the particular application, and without limitation to the scope of the invention, the experiment was carried out at the room temperature of 23 C. In  FIGS. 4 and 5  the biomacromolecule  130  concentration is taken 1 mg/ml of lysozyme, the pH quantity is 4.5 and the buffering agent is 50 mM sodium acetate. In  FIGS. 7-10  the additive is the salt NaCl, and the variable quantity  158 , taken to be the additive concentration  157 , is varied from 0 to 2.4 M in steps of 0.4 M. In  FIGS. 7 and 8  the assembly parameter  164  is taken to be the surface tension s. In  FIGS. 9 and 10  the assembly parameter  164  is the surface pressure p=s−s 0 , where s 0  is the surface tension of the solvent in the absence of biomacromolecules. For clarity, and without limitation to the scope of the invention, in  FIGS. 7-10  only three values of the additive concentration  157  are used to illustrate the second embodiment of the invention: 0.0 M NaCl  182 , 0.4 M NaCl  184  and 1.6 M NaCl  186 .  
         [0082]      FIG. 7  shows plots  191 ,  192 ,  193  of the assembly parameter  164  taken to be the surface tension s, against time  180 , for, respectively, the three additive concentrations  182 ,  184  and  186 , taking the additive to be the salt sodium chloride (NaCl). We see that the corresponding assembly parameters  164 , taken to be the surface tensions, remain unchanged when equilibrium times t eq    194 ,  195  and  196  are reached. Each equilibrium time t eq  corresponds to an equilibrium surface tension Seq.  
         [0083]      FIG. 8  shows plots  197 ,  198 ,  199  of the assembly parameter  164  against the square root of the time  180  for, respectively, the three additive concentrations  182 ,  184 ,  186 .  FIG. 9  shows plots  202 ,  204 ,  206  of the logarithm of the time derivative of the assembly parameter (in dp/dt)  164 , taken to be the surface pressure p, against the surface pressure p, for, respectively, the three additive concentrations  182 ,  184 ,  186 .  FIG. 10  illustrates time-dependence profile  220  of the macromolecule behavior against time  180 , which expression in the present application is given by ln(1−p/p eq ), where ln is the natural logarithm, p is the surface pressure and p eq  is an equilibrium surface pressure. The equilibrium surface pressure is given by p eq =s eq −s 0 , where s eq  is the equilibrium surface tension.  
         [0084]     In plots  212 ,  214 ,  216  of the configurational rearrangement expression  220  the equilibrium surface pressures p eq  are equal to the surface pressure evaluated, respectively, at the equilibrium times t eq    194 ,  195 ,  196  in  FIG. 7 .  
         [0085]     We notice that each plot in  FIGS. 8, 9  and  10  exhibits three line segments identifying the steps  142   144  and  146  in  FIGS. 4 and 5 : a first straight line segment  221  identifies the diffusion step  142 , a second straight line segment  222  identifies the penetration step  144 , and a third straight line segment  223  identifies the rearrangement step  146 . In  FIG. 10  the diffusion time t diff  is calculated by the inverse of the slope of the first straight line segment  221 , the penetration time t pen  is calculated by the inverse of the slope of the second straight line segment  222 , and the rearrangement time t arr  is calculated by the inverse of the slope of the third straight line segment  223 , so that the integration time is obtained as a sum of the last two, t int =t pen +t arr .  
         [0086]      FIG. 11  shows a plot  224  of the diffusion time t diff , a plot  226  of the penetration time t pen , a plot  228  of the rearrangement time t arr , and a plot  230  of the integration time t int =t pen +t arr , against the additive concentration  157 . In this experiment the additive concentration  157  is taken in a range from 0 to 2.4 M NaCl in steps of 0.4 M NaCl. A histogram in  FIG. 12  illustrates how the crystallization coefficients  232  relate to the aggregation boundary condition. The crystallization coefficients  232  k cryst =t int /t diff , are calculated for all pairs of obtained values for the integration time t int  and the diffusion time t diff , and are shown against the additive concentration  157  in the above range.  
         [0000]     Crystallization Window  
         [0087]     Subsequently crystallization experiments were carried out in order to quantify the aggregation boundary condition in terms of the crystallization coefficient k cryst    232  in  FIG. 12 . The experimental conditions used included the aforementioned range of the additive concentration  157  ( FIGS. 11 and 12 ) as well as the remaining applied experimental conditions referring, e.g. to pH and temperature. The biomacromolecule concentration used for the purpose of crystallization was allowed to vary. The crystallization results are classified in three domains  234  listed below and included in  FIG. 12 , where the crystallization coefficient  232  is shown against the additive concentration  157 .  
         [0088]     Domain A: The crystallization coefficient k cryst    232  is below approximately 4.0. Neither crystallization nor aggregation occurs in the entire range of the biomacromolecule concentrations used for crystallization, at the applied experimental conditions, e.g. additive concentration  157 .  
         [0089]     Domain B: The crystallization coefficient k cryst    232  is above approximately 4.0 and below approximately 8.5. Crystallization occurs for certain biomacromolecule concentrations, at the applied experimental conditions, e.g. the additive concentration  157 .  
         [0090]     Domain C: The crystallization coefficient k cryst    232  is above approximately 8.5. Only aggregation, but no crystallization, occurs at the applied experimental conditions, e.g. additive concentration  157 , regardless of the biomacromolecule concentrations used for crystallization.  
         [0091]     Thus the aggregation indicator  232  employed to define the aggregation boundary condition in  FIG. 12  lies below approximately 8.5, and the crystallization window is approximately given by the range 4.5&lt;k cryst &lt;8.5. An empirical result of our experiments is that in practice, a diffusion time that is short enough to enable crystallizing particles to approach one another and group together is reflected in a minimum value of k cryst =4 or 4.5. That means, that below about K cryst =4 or 4.5 crystallization is usually not observed. The value of the crystallization coefficient indicating the onset of amorphous aggregation is the upper bound, empirically found to lie around K cryst =8.5 or 9.  
       EXAMPLE 1  
     Experimental Result from Solution of the Protein Lysozyme in Sodium Chloride  
       [0092]     A combination of the crystal equilibrium condition in  FIG. 1A  or  FIG. 6  and the aggregation boundary condition in  FIG. 1B  or  FIG. 12  are used to illustrate how crystallization and aggregation conditions can be predicted. As regards the crystal equilibrium condition, the point  171  on the solubility curve  170  of  FIG. 1A  or  FIG. 6  corresponds to a pair of the additive (salt) concentration  157  that equals 1 M NaCl), and the biomacromolecule (lysozyme) concentration  156  (that equals 4 mg/ml). The crystal equilibrium condition is prescribing that crystallization can occur when the salt and lysozyme concentrations are taken above this pair of values. Thus the equilibrium lysozyme concentration equals 4 mg/ml when the additive concentration  157  is 1 M NaCl. The crystallization lysozyme concentration must therefore exceed 4 mg/ml.  
         [0093]     As regards the aggregation boundary condition,  FIG. 1B  or  FIG. 12  shows that when the salt concentration  157  is taken above around 1.9 M NaCl, no crystallization may be expected to occur, regardless of the lysozyme concentration used for crystallization; in other words any lysozyme concentration used for crystallization in this case has a very high probability of resulting in the amorphous aggregation. The crystallization coefficient k cryst    232  associated with the pair: salt concentration  157  equal to 1 M NaCl, lysozyme concentration  156  equal to 1 mg/ml in  FIG. 1B  or  FIG. 12  is approximately 4.8. Therefore, when the salt concentration is 1 M NaCl, it should be possible for the crystallization to take place. That means that it should be possible to find some range of lysozyme concentration to use for the purpose of successful crystallization. Indeed, the crystallization does occur when the lysozyme concentration used for crystallization in this case is above 20 mg/ml.  
       EXAMPLE 2  
     Experimental Result from Solution of Protein Lysozyme in Ammonium Sulfate  
       [0094]      FIG. 13  shows a plot  240  of the assembly parameter  164  taken to be the surface tension s, against time  180 , when the biomacromolecule solution  242  includes the protein lysozyme and the additive salt ammonium sulfate (NH 4 ) 2 SO 4 . The biomacromolecule concentration  156  is 1 mg/ml, and the additive concentration  157  is 2 M. This experiment resulted in the following values for the diffusion time, the penetration time, the rearrangement time, and the integration time: t diff =833.33 sec, t pen =2320 sec, t arr =5537 sec, t int =7857 sec. The corresponding value of the crystallization coefficient  232  k cryst  in  FIG. 12  was 9.43. According to the aggregation boundary condition established in the present method, crystallization experiments using the aforementioned lysozyme and ammonium sulfate concentrations should lead to the amorphous aggregation  128  in  FIG. 3 . Reports in the scientific literature confirm our result that lysozyme cannot be crystallized from the ammonium sulfate solution. However use of the method employing the second virial coefficient B 22  leads to the incorrect prediction that the crystallization  126  in  FIG. 2  of lysozyme from ammonium sulfate should be possible.  
       EXAMPLE 3  
     Experimental Result from Solution of Protein Concanavalin A in Ammonium Sulfate  
       [0095]      FIG. 14  shows a plot  244  of the assembly parameter  164  taken to be the surface tension s, against time  180 , when the biomacromolecule solution  246  includes the protein concanavalin A (about 23 kdalton) and the additive salt ammonium sulfate (NH 4 ) 2 SO 4 . The biomacromolecule concentration  156  is 0.1 mg/ml and the additive concentration  157  is 1 M. This experiment resulted in the following values for the diffusion time, the penetration time, the rearrangement time, and the integration time: t diff =277.78 sec, t pen =1436.78 sec, t arr =675.67 sec, t int =2112.45 sec. The corresponding value of the crystallization coefficient  232  k cryst  in  FIG. 12  was 7.605. According to the aggregation boundary condition established in the present method, crystallization experiments using the aforementioned concanavalin A and ammonium sulfate concentrations should lead to the crystallization  126  in  FIG. 2 . Reports in the scientific literature confirm our result that concanavalin A does indeed crystallize from the ammonium sulfate solution  246 . No predictions based on the method employing the second virial coefficient B 22  are available in the literature.  
       EXAMPLE 4  
     Experimental Result from Solution of Protein Bovine Serum Albumin in Ammonium Sulfate  
       [0096]      FIG. 15  shows plots  251 ,  252 ,  253 , of the assembly parameter  164  taken to be the surface tension s, against the time  180 . The biomacromolecule solution  250  includes the protein Bovine Serum Albumin (BSA, 66 kdalton) and the additive salt ammonium sulfate (NH 4 ) 2 SO 4 . The biomacromolecule concentrations  156  are, respectively, 0.01 mg/ml, 0.1 mg/ml 254 and 1.0 mg/ml, and the additive concentration  157  is 52% saturated ammonium sulfate. This experiment resulted in the following corresponding values of the crystallization coefficient  232  k cryst  in  FIG. 12 : 24.6, 24.8 and 23.8. According to the aggregation boundary condition established in the present method, crystallization experiments using the aforementioned BSA and ammonium sulfate concentrations should lead to the amorphous aggregation  128  in  FIG. 3 . Reports in the scientific literature however indicate that crystallization does indeed occur. This discrepancy is attributed to the tendency of BSA to dissociate and unfold on the surface  152  of the solution  120  in  FIG. 4 , i.e. in this case the solution/air interface. As a consequence of this tendency, the surface tension measurements prescribed in the present method offer an unreliable picture of the BSA crystallization window.  
       CONCLUSION  
       [0097]     The prediction of the crystallization window through the embodiments of the present invention features the crystal equilibrium condition, with reference to  FIGS. 1A and 6 , and the aggregation boundary condition, with reference to  FIG. 1B  and  FIG. 12 . Both embodiments are applicable for biomacromolecules that are not prone to severe unfolding at the surface  152  of the solution  120  ( FIG. 4 ). The crystal equilibrium condition follows from the biomacromolecule solubility curve  170  ( FIG. 6 ), and defines the lower bound of the biomacromolecule concentration to be used for crystallization.  
         [0098]     The boundary aggregation condition refers to the solvent experimental parameters. It is applicable when the biomacromolecule solutions used to determine the crystallization coefficient k cryst    232  ( FIG. 12 ) are dilute. The results are summarized for various experiments in Table 1, where the employed biomacromolecules are proteins. The first column lists the protein concentrations  156  in  FIGS. 7-15  used for predicting the crystallization window by determining the aggregation boundary condition. The protein concentrations used for the crystallization in the case of lysozyme are allowed to vary in a range 5-100 mg/ml (not reported in Table 1). The second column lists the salt concentration  157  ( FIGS. 7-15 ) used for determining the aggregation boundary condition, and also featuring in the crystallization experiment. The third column lists the pH, and the fourth column lists the resulting crystallization coefficient  232  k cryst  ( FIG. 12 ). The fifth column lists the outcome of the crystallization experiments in terms of the domains A, B or C, as predicted by the aggregation indicator of the present method. These predictions are compared with the predictions obtained from the method used in the prior art employing the second virial coefficient B 22 . Both methods are compared with experimental results in the last two columns of Table 1, whenever such results are available.  
                                                                     TABLE 1                       Protein,   Salt,           Crystallization               concentration   concentration           result,   B 22     Surface tension       (mg/ml)   (M)   pH   k cryst     domain ( FIG. 12 )   prediction   prediction                                lysozyme,   sodium chloride,   4.5   0.8   solution, A   correct   correct       1.0   &lt;0.3       lysozyme,   sodium chloride,   4.5   4-7    crystal, B   correct   correct       1.0   0.3-2.0       lysozyme,   sodium chloride,   4.5   8-12   aggregation, C   correct   correct       1.0   &gt;2.0       lysozyme,   ammonium sulfate,   4.5   9.5   aggregation, C   incorrect   correct       1.0   2.0       concalavine A,   ammonium sulfate,   7.0   7.6   crystal, B   unavailable   correct       0.1   1.0       BSA,   ammonium sulfate,   6.2   24   aggregation, C   unavailable   incorrect       0.01, 0.1, 1.0   52% saturated                  
 
         [0099]     We see from Table 1 that the aggregation boundary condition in  FIG. 12  prescribes that when the crystallization coefficient  232  is larger than approximately 8.5, the biomacromolecule has a high probability to undergo the amorphous aggregation  128  in  FIG. 3  at the employed experimental conditions involving, e.g., specifications of solvent, additive, their concentrations or the pH. Hence the aggregation indicator defines successfully a relatively narrow crystallization window 4.5&lt;k cryst &lt;8.5, or a broader crystallization window 4.0&lt;k cryst &lt;9.0, for biomacromolecules, with the following provisions: first that the biomacromolecules are able to maintain their conformations at the solution/air interface; and, second, relatively low biomacromolecule concentrations, e.g. not exceeding approximately 1 mg/ml, are used in the prescribed experiments to determine the aggregation boundary condition.