## Abstract

We report a real-time measurement method of the solution concentration variation during the growth of protein-lysozyme crystals based on digital holographic interferometry. A series of holograms containing the information of the solution concentration variation in the whole crystallization process is recorded by CCD. Based on the principle of double-exposure holographic interferometry and the relationship between the phase difference of the reconstructed object wave and the solution concentration, the solution concentration variation with time for arbitrary point in the solution can be obtained, and then the two-dimensional concentration distribution of the solution during crystallization process can also be figured out under the precondition which the refractive index is constant through the light propagation direction. The experimental results turns out that it is feasible to in situ, full-field and real-time monitor the crystal growth process by using this method.

©2012 Optical Society of America

## 1. Introduction

In recent decades, defect-free protein crystals with a high degree of purity are important to the drug design, biochemical and biophysical investigations, etc. The crystallization processes investigation is the basis of the theoretical study about crystals growth and the preparation of the required high-quality crystals. As many phenomena in the crystallization process such as mass transfer can be reflected by the solution concentration variation [1], it is crucial to measure the change of solution concentration during the crystallization process. In recent years, several optical methods are used for monitoring the solution concentration variation such as schlieren technique [2], shadowgraph technique [3], holographic interferometry [4] electronic speckle pattern interferometry (ESPI) [5] and interferometry [6, 7]. By using schlieren and shadowgraph techniques, the concentration trend can be measured qualitatively. However, it is difficult to gain quantitative information. Since traditional holographic interferometry needs wet chemical process and other time-consuming procedures, it can’t be used for real-time measurement the solution concentration variation. ESPI can be used under incoherent lighting conditions, yet the contrast of interference fringes is low due to the speckle noise itself, and thus reduces the resolution of the measurements. Currently, interferometry is one of the most widely used approaches. It can be used to monitor the solution concentration variation during crystallization process by counting the shift number of interference fringes within the sampling window. To further increase the measurement sensitivity, phase shifting technique has been incorporated into the interferometry [8, 9]. A drawback of this method, however, is the complexity of the optical path, the uncertainty of the phase-shift value [10] and non-real-time. In addition, the existing measurement methods mainly focus on measuring the solution concentration variation in the sampling window without crystals. New techniques that can effectively detect the solution concentration variation during the crystallization process are still desirable for studying protein crystallization process quantitatively.

In recent years, digital holographic interferometry has been widely used in deformation measurement, flow field visualization, microscopy, temperature measurement, physics process monitoring etc, owing to its non-destructive, high precision and full-field measurement [11–18]. In this paper, we propose a method to achieve the real-time and full-field monitoring of the solution concentration during the crystallization process based on digital holographic interferometry. For measuring the concentration variation during the crystallization process, the changing procedure of the two-dimensional concentration distribution of the solution with time can be recorded in form of holograms using CCD continuously. By numerically simulating the diffraction of the digital holograms, the information of the solution concentration variation during crystallization can be reconstructed by computer. Under the condition of constant temperature, the phase difference between the reconstructed object waves in different states is only a function of the solution concentration. Therefore, the two-dimensional full-field solution concentration distribution and the concentration-time curve of arbitrary point in solution can be obtained by detecting the corresponding phase difference. Compared with other methods, this method is very convenient and sensitive enough.

## 2. Principles

According to the principle of digital holographic interferometry, the quantitative phase information of the object wave can be obtained by holographic reconstruction [15, 19]. That means the solution concentration change causing phase distribution variation of the object wavefront can be measured by digital holographic interferometry. Taking *t* as the growth time of the crystals in the crystallization process, *φ _{ot}*(

*x*,

*y*) as the phase distribution of the object wave at time

*t*, then the phase difference of the reconstructed object wavefront corresponding to the solution at time

*t*relative to the initial moment (

*t*= 0min) can be expressed as

As the thickness of the solution which object wave passing through is very small (0.2cm) in this experiment, the refractive index of the solution in the direction of the optical path (*z* axis) can be viewed as constant, thus we can assume that the thickness of the solution at (*x*,*y*) along the optical pathway is *d* (*x*,*y*), and *n*_{0} and *n _{t}*(

*x*,

*y*) are the refractive index of the solution at initial time and at time

*t*, respectively. Then we have

*C*

_{0}(

*x*,

*y*) and

*T*

_{0}(

*x*,

*y*) are the solution concentration and the temperature at initial time,

*C*(

_{t}*x*,

*y*) and

*T*(

_{t}*x*,

*y*) are the solution concentration and the temperature at time

*t*, (∂

*n*/∂

_{s}*T*)

*and (∂*

_{C}*n*/∂

_{s}*C*)

*are the dependence of the refractive index of the solution on temperature and concentration, respectively.*

_{T}Under the condition of the constant temperature, Eq. (3) can be simplified as

## 3. Experimental setup

Figure 1
depicts the experimental setup for measuring the solution concentration based on digital holographic interferometry. A thin laser beam with *λ =* 532nm is split into two parts by a beam splitter BS_{1}. Both the reflected and transmitted beams are expanded, filtered and collimated via microscope objective MO, pinhole PH and lens L to generate the reference and object waves, respectively. The object wave passes through a 1.26cm × 0.2cm × 2.14cm growth cell and then interferes with the reference wave through a beam splitter BS_{2}. The growth cell is imaged on CCD by telecentric lens (TL). The CCD is a black and white type with 1626_{H} × 1326_{V} pixels and the pixel size is 4.4μm × 4.4μm. The distance between TL and the growth cell is 150mm. Batch method is used to crystals growth in the experiment. The solution is prepared by mixing 150μL lysozyme solution (28mg/ml) and 150μL NaCl Solution (35mg/ml). The mixed solution (PH = 4.6) is injected in the growth cell and sealed with a lid immediately. The temperature is controlled at 17.5°C.

## 4. Experimental results and analysis

During the crystal growth, 4752 holograms are recorded by CCD at the frequency of 1 frame per 1min. According to the principle of the double-exposure holographic interferometry, setting the first hologram corresponding to the initial state, we obtained the two-dimensional wrapped phase difference distributions of the solution region. as show in Figs. 2(a)
–2(h) with *t =* 200min, 1000min, 1550min, 2350min, 3050min, 3700min, 4200min, 4750min, respectively. In order to observe the phase evolution process more clearly, we displayed the process in the form of a movie (Media 1).

From Figs. 2(a) and 2(b), we can see that the phase variation is almost identical in different solution regions before crystals appear. In Fig. 2(c), the appearance of a few crystals at the solution bottom causes the emergence of phase mutation. By comparing Figs. 2(d)–2(h), we can see that there only adds a few new crystals in the solution from the time 2350min and the quantities of the crystals at the solution top are much more than that at the bottom. The number of the wrapped fringes in the solution region increase with the crystal size proportionally, as shown in Fig. 2(e)–2(h).

In order to observe how the solution concentration varies with time in different regions, we choose three points A, B, C at top, middle and bottom in the solution regions, respectively, as shown in Fig. 3(a)
. Considering that the crystallization process lasts for a very long time (4752min), several factors which leads to inaccurateness of the experimental results should be considered, such as the vibration caused by the Lens. For the purpose of removing the influence of the environmental noise on the phase difference, we choose an area with 10 × 10 pixels in the growth cell marked by a red rectangle D which doesn’t include the solution, and subtract the average phase of the area from the phase difference in point A (B, C), as shown in Fig. 2(a). According to the phase unwrapping algorithm and the Eq. (5), the value of (∂*n _{s}*/∂

*C*)

*is taken as 12.16 × 10*

_{T}^{−4}mg/ml, and the solution concentration variation curves during 4752 minutes at the three points are obtained, respectively, as shown in Fig. 3(a). Figure 3(b) shows the fitting results corresponding to Fig. 3(a). It is obtained by 5th degree polynomial. From Eq. (5), it can be known that the solution concentration accuracy of this method mainly depends on the precision of phase, and the phase measurement accuracy in digital holographic interferometry for our case is ~0.25° [20]. Hence we can figure out the final concentration result with a precision of 1.47 × 10

^{−4}mg/ml.

Comparing the three solution concentration curves, we can see that all of them decrease very smoothly during the nucleation process (in first 1800 minutes). The reason is that the nucleation process just consumes a small quantity of protein molecules in the three regions. Then the speed of the protein consumption accelerates at the crystals growth stage (from about 2400min). More protein molecules at the top solution are used to incorporate into the crystals, which induces the differences among the three solution concentration curves increase with time, as shown in Fig. 3(a).

By utilizing the phase unwrapping algorithm and the concentration value of point A (B, C), we can receive a series of two-dimensional solution concentration distributions in the crystals growth process.

Figure 4
shows the two-dimensional solution concentration distributions of the area marked by a black dashed rectangle in Fig. 2(a) in different time. The area contains 600_{H} × 510_{V} pixels (about 7.92mm × 6.72mm). As shown in Figs. 4(a) and 4(b), the solution concentration distributions almost have the same downtrends in the full-field solution region, and the solution concentration variation is very small in the initial stage (i.e. in the initial 1000min). This is because the rate of the protein molecules consumption is slow and the consumption of protein molecules is nearly similar at the beginning of nucleation process. With the reaction carried through (at about 1550min), several crystals appeared at the solution bottom and the concentration at the solution bottom becomes smaller than that at the top, as shown in Fig. 4(c). The reason is that the nucleation process occurs earlier, and consumes more protein molecules at the solution bottom because the interface of the cell bottom is more helpful to the crystal nucleation. From Figs. 4(d) to 4(h), we calculate out that the solution concentration gradient increases between the top and bottom part with the crystals growth (between 2350min and 4750min). Comparing Figs. 4(d) with 4(h), the solution concentration gradient is about 0.6mg/ml at 2350min, and 2mg/ml at 4750min. This phenomenon is due to the reason that the more the crystals in the solution top are, the faster the consumption speed of the protein molecules is.

It is noticed that here we record 1 frame hologram per 1 minute because the speed of the solution concentration variation is very slow. Actually, we can capture a series of holograms in video speed of 25 frames per second or higher, which only depends on the changing speed of the object field and the acquisition speed of the CCD. That is to say, by means of appropriate acquisition speed of the CCD, a real-time measurement can be achieved with this method.

## 5. Conclusions

In summary, the phase change of the object wavefront passing through a crystal growth cell during the crystallization process is real-time monitored by digital holographic interfermetry. Based on the relationship between the phase difference of the reconstructed object wavefronts in different states and the solution concentration variation, we obtained the two-dimensional solution concentration distribution during the crystal growth. To our knowledge, it is difficult to measure by other methods under such growth environment, as the crystals suspend dispersedly in the sampling window. By analyzing the solution concentration curve in different regions during the crystallization process, we discussed the influences of the crystals distribution on the solution concentration variation in the process. It is noticed that the proposed method provides a robust real-time and full-field method to measure the crystallization process precisely.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61077008 and 61127011 and the Northwestern Polytechnical University Foundation for Fundamental Research under Grant No.JC20100237.

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