Modeling
BIOMANUFACTURING OPTIMIZATION USING BOX-BEHNKEN DESIGN
ITERATION №1
The goal of iteration one was to find the gold:silver salt ratio for chemical synthesis of bimetallic nanoparticles with ascorbic acid
ITERATION №2
The goal of iteration two was to find the optimal gold and silver salts concentration, pH and temperature for biological synthesis of bimetallic nanoparticles with a cell-free system
Overview
We have provided an elaborate summary of our modeling experiments and design. Read further below for detailed information on the model.
What?
Specific parameters, such as temperature, pH and gold and silver concentration, significantly impact the morphology of the nanoparticles formed, which impacts the absorption spectrum. We built a model to predict how these synthesis conditions influence the absorption spectrum of nanoparticles.
How?
We performed nanoparticle synthesis with varying gold and silver concentration, pH, and temperature and measured the absorption spectrum of the nanoparticles. To this data we applied a model called Box-Behnken Design (BBD) and obtained equations describing the absorbance at different wavelengths as a function of the varied parameters.
Why?
Photothermal therapy (PTT) is more effective at ablating tumor cells when nanoparticles are used that have a high near-infrared red absorption1. From our model we obtained the synthesis conditions that maximize the absorbance at 800 nm (A800 nm) of our nanoparticles, making them more suitable for PTT.
Short description of BBD
Fig. 1 gives a visual representation of how BBD works. \(A\), \(B\), and \(C\) represent the independent factors (in our case the different synthesis conditions) and \(y\) the dependent factors (in our case, the absorbance at a certain wavelength). BBD can be applied if the value of \(y\) is measured at certain combinations of \(A\), \(B\), and \(C\). BBD then produces a quadratic equation that describes \(y\) as a function of \(A\), \(B\), and \(C\). This equation is subsequently used for optimization.
Setup of our modeling
We performed our modeling in two iterations.
Iteration one
In the first iteration, we used ascorbic acid to synthesize nanoparticles at different values of the pH, gold and silver concentration. By feeding this data into BBD we obtained an equation expressing the A800 nm in terms of pH, gold concentration and silver concentration. We used this equation to perform optimization. We wanted to find the gold:silver ratio that maximizes the A800 nm. To achieve this, we implemented two optimization criteria: (1) that the A800 nm should be maximized while (2) the pH is kept at 7.3, since this is the pH of the medium used in the second iteration and in all other experiments. This gave us a gold:silver ratio of 1.85:1, which was used in subsequent experiments.
Iteration two
In the second iteration, a cell-free system was used and the gold concentration, pH, and temperature were varied. The gold:silver ratio was kept at 1.85:1. Experiments were performed with WT Escherichia coli (E.coli) supernatant with WT E. coli lysate, WT E. coli supernatant, and ddH2O. From this, we found the optimal gold and silver concentration, pH, and temperature to perform nanoparticle synthesis for PTT in a cell-free system.
Results
The results of our modeling are visualized in Fig. 3 and Fig. 4. Adjust the parameters yourself to see the effect on the absorption spectrum.
The shape of the peak gets noticeably sharper when the pH goes from pH 5 to pH 7 (Fig. 3). At pH 9 the absorbance is so low, that the model predicts it is negative for certain wavelengths. This is a fault of the model, since there was no negative absorbance measured during the experiments. At a gold concentration of 0.25 mM, the peak is around 600 nm. However, for lower values the peak is around 700 nm. Interestingly, the absorption spectrum looks similar when the silver concentration is low (around 0.05 mM) and when it's high (around 0.15 mM). The largest change in the spectrum occurs when the silver concentration shifts towards a concentration of 0.10 mM. At that point, the absorbance at 700 nm increases.
Fig. 4 shows that the solutions' absorbance is affected differently by the synthesis conditions than in iteration one. In iteration one, there was a distinct shift of the peak when synthesis conditions changed. However, in Fig. 4, we observe that a change in parameters mainly affects the height of the graph, but not the shape. The exception to this is the effect of pH on the nanoparticles produced using supernatant. Absorbance above 850 nm increases as the pH increases, while the opposite holds for the rest of the absorbance values.
Implementation
The gold:silver ratio obtained in iteration one has been implemented in our other experiments (see Engineering Cycle). In further production of our nanoparticles, the parameter values obtained at iteration two would be used.
A strength of our model is that we can maximize the A800 nm while applying any other necessary criteria. This is useful in implementation and upscaling. For example, it could be desirable to minimize energy usage in a nanoparticle manufacturing facility while still maximizing A800 nm. Then the optimization criteria can be set as (1) maximize the A800 nm and (2) minimize the temperature. Another example is minimizing costs by restricting the concentration of gold and silver salts used. Our model is therefore valuable in upscaling to ensure the economic viability of Binanox.
Conclusion
By employing our model we gained insight into how synthesis conditions affect the absorption spectrum of nanoparticles and found the optimal synthesis conditions for our biomanufacturing process and potential future implementations.
Contribution
At the bottom of our page, we provide a step by step outline for future teams on how to apply BBD in their projects.
Box-Behnken Design
A one-factor-at-a-time approach is primarily used in biology to determine the influence of parameters on an observation. However, the interplay between several components can be overlooked if only one factor is considered at a time. For example, high pH and low temperature may individually benefit cell growth, while their combination may be inhibiting. Therefore, obtaining optimal conditions requires investigating such interactions. However, when the number of factors increases, such an investigation becomes too labor-intensive.
Responses Surface Methodology (RSM) is a method for reducing the number of experiments necessary for obtaining statistically relevant results. This set of statistical techniques allows for the analysis of independent and dependent variables5. Box-Behnken Design (BBD) is a type of RSM design widely applied in research for optimization problems.
BBD is more efficient than other forms of RSM, such as Central Composite Design (CCD)6. BBD helps reduce the number of experiments required to identify the optimum. With this, BBD allows for optimization of the dependent variable of interest by measuring the dependent variable at predetermined combinations of the independent variables. In our case, the absorbance is the dependent variable. A visual representation of the general principle of BBD is given in Fig. 5 and described below.
A, B, and C represent the three independent variables. The values -1 and 1 illustrate the value range of these variables. These values are called levels. The third level is level 0. It annotates the central value of the independent variable. The central value is hypothesized to be optimal. Values -1 and 1 represent the minimum and the maximum of the range. By the rules of BBD, these values must be at equal distances from the central value. BBD can only predict the value of the dependent variable within the tested ranges of the independent variable. Therefore, it is important to choose the ranges of the independent variables such that their optimal value does not lie outside the ranges.
An experiment is set up to discover the optimum pH, temperature, and solar radiation for E. coli growth using BBD. Since 37°C is the optimal temperature for E. coli7, it is reasonable to choose this value as the middle value. The temperature range of interest is between 21°C and 49°C degrees, since E. coli's growth is inhibited outside these ranges8. Therefore, a suitable range for the temperature could be 37±12°C. Table 1 gives the temperature values that correspond to the three levels.
| Level | Temperature (°C) |
| -1 | 25 |
| 0 | 37 |
| 1 | 49 |
The colored circles in Fig. 5 represent the combinations of variables at which the optimizable parameter is measured. Not every place on the box has a circle, which means that not every combination needs to be tested. For example, the combination (A, B, C) = (-1,-1,0) is tested, but (A,B,C) = (-1,-1,-1) is not. The central point is the point at which all parameters are at value zero. It is measured multiple times to ensure the model's accuracy. The number of experiments that have to be performed for BBD is given by the following formula:
$$N=2k(k-1)+C_o$$
Hereby \(k\) is the number of factors and \(C_o\) is the number of times the central point is measured. After obtaining the measurements, a quadratic equation is fit to the results. This equation is used to optimize the dependent variable. The equation obtained by applying BBD is of the following form:
$$y = c_1A + c_2B + c_3C + c_4 AB + c_5 AC + c_6 BC + c_7 A^2 + c_8 B^2 + c_9 C^2$$
Hereby, \(y\) is the dependent variable and \(A\), \(B\), and \(C\) are the dependent variables. The terms \(AB\), \(AC\), and \(BC\) signify the effect of the interaction of two independent variables on the dependent variable.
Setup of the modeling
Fig. 6 | Flowchart of the steps undertaken to apply BBD.
The following steps were undertaken to build the model:
- We determined which independent factors would be tested. We researched which synthesis conditions affect the absorption spectrum of nanoparticles.
- Since BBD can only find the optimal conditions within the tested parameter ranges, we performed preliminary experiments in which we varied the value of each parameter individually. From the results we concluded which parameter ranges we should use. For some parameters, we obtained the optimal range from literature.
- After obtaining the ranges we performed nanoparticle synthesis under certain combinations of the parameters. We measured the absorption spectra of the synthesized particles.
- This data was fed into the software Design Expert by Stat-Ease. We analyzed the data and applied BBD to fit a quadratic equation to the results.
- Using the numerical optimization tool in Design Expert, we maximized the A800 nm and found the optimal conditions. Design Expert gives multiple solutions for the maximization and selects one as the recommended ones. We used the recommended solutions to obtain our optimal conditions.
Iteration one: Nanoparticle synthesis with ascorbic acid
Determining the experimental parameters and their ranges
Gold and silver salt concentrations, pH, and temperature have shown to have a significant effect on the absorption spectrum of nanoparticles2,3,4. Therefore, they were chosen to be the independent variables in our model. As previously mentioned, the central value of each parameter is the hypothetical optimal value for nanoparticle synthesis. Furthermore, BBD can only find optimal conditions within the tested parameter ranges. Hence, we performed preliminary experiments to find the central value and range for each parameter.
Gold and silver salt concentrations range is 0.20±0.05 mM and 0.10±0.05 mM respectively
Gold (HAuCl4) and silver (AgNO3) salt concentrations were varied in ddH2O with the addition of ascorbic acid. Details on the experimental procedure can be found in this protocol. Fig. 7 shows how this affected the A800 nm. The highest average absorbance was measured with 0.20 mM HAuCl4 and 0.10 mM AgNO3; therefore, these values were chosen as the central values for BBD. At the same time, the absorbance was high at 0.05 mM and 0.15 mM AgNO3, so we decided to make a range of 0.20±0.05 mM for HAuCl4 and 0.10±0.05 mM for AgNO3.
Fig. 7 | Effect of HAuCl4:AgNO3 ratio on the A800 nm of nanoparticles produced using ascorbic acid. The average of three chemical replicates with the confidence interval is shown. The average A800 nm is highest for a ratio of 0.20 mM HAuCl4 and 0.10 mM AgNO3.
The pH range is 7±2.
To bridge the gap between chemical and biological synthesis, the reduction with ascorbic acid was conducted at pH 7 instead of pH 3, as done in the reference paper 9. We presumed that E. coli's reducing enzymes function optimally at neutral pH, as it is the optimal pH for E. coli growth10. However, as this methodology deviates from the original paper, the ascorbic acid experiments were repeated at a biologically relevant pH to obtain an optimal gold:silver salt ratio for this pH range. pH 9 was chosen as an upper bound, since E. coli cannot grow above this pH 11. The final pH range was 7±2.
The parameter ranges with the central point we used for the modeling experiments are shown in Table 2. Our model assumes that our optimal conditions are within these ranges.
| Level | |||
| Variable | -1 | 0 | 1 |
| pH | 5 | 7 | 9 |
| [HAuCl4] (mM) | 0.15 | 0.2 | 0.25 |
| [AgNO3] (mM) | 0.5 | 0.1 | 0.15 |
Experimental procedure
As mentioned before, certain combinations of parameters have to be tested to apply BBD. Table 3 shows the different combinations tested for iteration one. In the first iteration, we used ascorbic acid for the nanoparticle synthesis. The synthesis-method consists of adding HAuCl4, AgNO3 and ascorbic acid to ddH2O. The description of our methods can be found in this protocol. For all 15 synthesis conditions we used, we measured the absorption spectrum of both the ascorbic acid-produced nanoparticles and the control, which was made up of gold and silver salts added to ddH2O. To distinguish between these two reduction solutions, we will refer to the solution that includes ascorbic acid as the it1-AA solution and to the control as the it1-control solution.
Solutions used in iteration one for nanoparticle synthesis:
it1-AA solution: ddH2O + ascorbic acid + gold and silver salts
it1-control solution: ddH2O + gold and silver salts
| Run | pH | [HAuCl4] (mM) | [AgNO3] (mM) |
| 1 | 5 | 0.15 | 0.10 |
| 2 | 9 | 0.15 | 0.10 |
| 3 | 5 | 0.25 | 0.10 |
| 4 | 9 | 0.25 | 0.10 |
| 5 | 5 | 0.20 | 0.05 |
| 6 | 9 | 0.20 | 0.05 |
| 7 | 5 | 0.20 | 0.15 |
| 8 | 9 | 0.20 | 0.15 |
| 9 | 7 | 0.15 | 0.05 |
| 10 | 7 | 0.25 | 0.05 |
| 11 | 7 | 0.15 | 0.15 |
| 12 | 7 | 0.25 | 0.15 |
| 13 | 7 | 0.20 | 0.10 |
| 14 | 7 | 0.20 | 0.10 |
| 15 | 7 | 0.20 | 0.10 |
The datatables with the results and the graphs can be found in the dropdown.
Table 4 | The it1-AA's absorbance between 450 and 1100nm for the different synthesis conditions. Download Table 4 here
Table 5 | The it1-control’s absorbance between 450 and 1100nm for the different synthesis conditions. Download Table 5 here
Data analysis
The Design Expert software from Stat-Ease was used to analyze the data gathered in our experiments (Table 4 and 5) and to maximize the A800 nm. The measured absorbance value was inputted for each combination of tested factors. The data was analyzed and a quadratic equation was fit to these input values. This equation was used to maximize the A800 nm.
Fig. 8 | An overview of the steps taken to create an accurate model
To build an accurate model of our data, we first ensured that the data is normally distributed. Data needs to be normally distributed before applying statistical tools, e.g., Analysis of Variance (ANOVA). A Box-Cox transformation was applied to achieve a normal distribution. This is a power transformation of the form yλ, with y the dependent variable and λ between -5 and 5 12. Table 6 gives common types of transformations and the corresponding value of λ.
The appropriate transformation can be chosen by generating a Box-Cox plot. Fig. 9 is an example of the Box-Cox plot of it1-AA’s A800 nm data. The green line represents the best λ and has a value of -0.25. It is visible in Table 8 that no transformation corresponds with λ = -0.25. In that case, the best transformation is the one, for which the λ is the closest to -0.25. In this case, that is λ = 0 (the logarithmic transformation) or λ = -0.5 (the inverse square root transformation).
| λ | Transformation |
| 2 | Squared |
| 1 | No transformation |
| 0.5 | Square root |
| 0 | Logarithmic |
| -0.5 | Inverse square root |
| 1 | Inverse |
| 2 | Inverse square |
After we applied a transformation, we ensured that the data was normally distributed by generating a normal probability plot of the data 13. If data follows a normal distribution, the data points form a straight line on the plot.
After we found the suitable transformation for each solution we chose the equation that fits the data. Design Expert recommended the best-fitting equation out of three:
|
Linear: of the form \(y = c_1A + c_2B + c_3C\) 2FI: of the form \(y = c_1A + c_2B + c_3C + c_4 AB + c_5 AC + c_6 BC\) Quadratic: of the form \(y = c_1A + c_2B + c_3C + c_4 AB + c_5 AC + c_6 BC + c_7 A^2 + c_8 B^2 + c_9 C^2\) |
Hereby, \(A\), \(B\), and \(C\) represent the independent factors, and \(c_1,\ldots, c_9\) are constants. In choosing the equation, the recommendation of Design Expert was followed. Then, the p-value of every term in the equation was examined. If the p-value was above 0.05, the term was removed. For example, for the A800 nm data of the it1-control solution, only the pH and the pH2 were significant terms. Thus all other terms were removed and the model looked as following:
The steps above were applied to the A800 nm of each solution. As mentioned above, not only the A800 nm was modeled, but also the absorbance for other wavelengths, such as A450 nm for example. Instead of undertaking the above-mentioned steps for each wavelength, we modeled every wavelength in the same way as we modeled the A800 nm. For instance, the A800 nm data of the it1-control was transformed with an inverse square root transformation and the model terms included only the pH and pH2. That means the data for all other wavelengths of that solution, were also transformed with an inverse square root transformation, and their model terms were made to only include the pH and pH2. We did this to avoid overfitting the model.
Table 7 summarizes which transformations were performed for the A800 nm data of it1-AA and it1-control. The full explanation behind these choices can be found in the dropdown below.
| Solution | Transformation | Model of the form |
| It1-AA | No transformation | \(A_{800 nm} = c_{AA_1} \text{pH} + c_{AA_2} \text{[HAuCl}_4] + c_{AA_3}\text{Temperature} + c_{AA_4} \text{pH}^2 + c_{AA_5} \text{[HAuCl}_4] ^2 + c_{AA_6}\text{Temperature}^2\) |
| It1-control | Inverse square root | \(\frac{1}{\sqrt{A_{800 nm}}}= c_{control_1} \text{pH} + c_{control_2} \text{pH}^2\) |
Table 8 provides the p-values. They are all below 0.05, indicating a significant relationship between the variables.
| Solution | p-value |
| It1-AA | 10-4 |
| It1-control | 10-4 |
All the steps undertaken to find a correct model for each solution are explained here.
Creating a model of the It1-AA A800 nm data.
Fig. 10 shows the Box-Cox plot for the it1-AA data of the A800 nm. The green line represents the best λ with a value of -0.25. Based on this, the software recommended performing an inverse square root transformation. However, the logarithmic transformation was also considered since its λ is close to -0.25. No transformation was also considered.
Fig. 10 | Box-Cox plot for the It1-AA A800 nm data. The green line represents the best λ (-0.25). The red lines represent the confidence interval (-0.52, 0.16), and the blue line represents the current transformation (1).
For all three transformations, the quadratic model was recommended. Although our software recommended inverse square root transformation, it gave inaccurate predictions for the A800 nm, suggesting absorbance as high as 1219 could be achieved. Therefore, only the logarithmic transformation and no transformation were considered. Fig. 11 shows the normal probability plots of the data without a transformation and with a logarithmic transformation. In both plots the data points follow the red line closely, which suggests that both transformations work well for this data. We therefore concluded that it was unnecessary to perform a transformation, since the data already follows a normal distribution.
Fig. 11 | The normal probability plot of the It1-AA A800 nm data. Left: no transformation. Right: logarithmic transformation.
What do the terms in the ANOVA table mean?
The sum of squares gives an indication of the variance of the data from the mean.
df stands for degrees of freedom. It is the number of independent pieces of information used to calculate a statistic. The mean square is the sum of squares divided by the degrees of freedom. The F-value is the variation between sample means divided by the variance within the sample. The p-value describes the likelihood of obtaining the data by random chance. Table 9 presents the ANOVA table of the model.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
1.52 |
9 |
0.169 |
11.31 |
7.8*10-3 |
| A-pH |
0.8366 |
1 |
0.8366 |
55.97 |
7*10-4 |
| B-Silver |
0.0126 |
1 |
0.0126 |
0.8431 |
0.4006 |
| C-Gold |
0.0262 |
1 |
0.0262 |
1.75 |
0.2431 |
| AB |
0.0017 |
1 |
0.0017 |
0.1138 |
0.7495 |
| AC |
0.0035 |
1 |
0.0035 |
0.2309 |
0.6511 |
| BC |
0.0056 |
1 |
0.0056 |
0.3713 |
0.5689 |
| A² |
0.4252 |
1 |
0.4252 |
28.45 |
0.0031 |
| B² |
0.1364 |
1 |
0.1364 |
9.13 |
0.0294 |
| C² |
0.1593 |
1 |
0.1593 |
10.66 |
0.0223 |
| Residual |
0.0747 |
5 |
0.0149 |
The terms AB, AC, and BC have p-values higher than 0.05. Therefore, we chose to remove them from the model. The impact of this is visible in Table 10. After removing the terms, the p-value of the model dropped.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
1.51 |
6 |
0.2518 |
23.57 |
1*10-4 |
| A-pH |
0.8366 |
1 |
0.8366 |
78.33 |
0.0001 |
| B-Silver |
0.0126 |
1 |
0.0126 |
1.18 |
0.309 |
| C-Gold |
0.0262 |
1 |
0.0262 |
2.45 |
0.1562 |
| A² |
0.4252 |
1 |
0.4252 |
39.81 |
0.0002 |
| B² |
0.1364 |
1 |
0.1364 |
12.77 |
0.0073 |
| C² |
0.1593 |
1 |
0.1593 |
14.92 |
0.0048 |
| Residual |
0.0854 |
8 |
0.0107 |
Creating a model of the it1-control A800 nm data.
The Box-Cox plot of the it1-control A800 nm data is presented in Fig. 12. Based on the value of λ, an inverse square root transformation was recommended.
Fig. 12 | Box-Cox plot for the it1-control A800 nm data. The green line represents the best λ (-0.67). The red lines represent the confidence interval (-1.77, 0.8), and the blue line represents the current transformation (1).
The normal probability plot is given in Fig. 13. Although the data points don’t exactly follow the red line, there are no strong outliers. Therefore, it was decided not to adjust the transformation.
Fig. 13 | The normal probability plot of the it1-control A800 nm data after applying an inverse square root transformation.
After having applied the transformation, the model recommended a quadratic model. However, in the ANOVA table (Table 11), we see that the only significant term is A2. A2 represents the pH2. Therefore, only the pH substantially impacts the absorption spectrum of our control. This is a reasonable result since the volume of the buffer added to adjust the pH was greater than the maximum volume of silver and gold salts added. Therefore, we decided to adjust our model such that the pH2 and pH would be the only terms.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
69.29 |
9 |
7.7 |
3.58 |
0.0868 |
| A-pH |
27.89 |
1 |
27.89 |
12.98 |
0.0155 |
| B-Silver |
0.1229 |
1 |
0.1229 |
0.0572 |
0.8205 |
| C-Gold |
0.4394 |
1 |
0.4394 |
0.2045 |
0.6701 |
| AB |
0.0201 |
1 |
0.0201 |
0.0094 |
0.9267 |
| AC |
0.3448 |
1 |
0.3448 |
0.1604 |
0.7053 |
| BC |
1.01 |
1 |
1.01 |
0.4698 |
0.5236 |
| A² |
33.09 |
1 |
33.09 |
15.4 |
0.0111 |
| B² |
1.48 |
1 |
1.48 |
0.6887 |
0.4444 |
| C² |
3.57 |
1 |
3.57 |
1.66 |
0.2539 |
| Residual |
10.75 |
5 |
2.15 |
We see now that for the adjusted model, the p-value is below 0.05 (Table 12).
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
61.91 |
2 |
30.96 |
20.5 |
0.0001 |
| A-pH |
27.89 |
1 |
27.89 |
18.47 |
0.001 |
| A² |
34.02 |
1 |
34.02 |
22.53 |
0.0005 |
| Residual |
18.12 |
12 |
1.51 |
Results
The maximization of A800 nm of the it1-AA was set as an optimization criterion in Design Expert. This resulted in pH 6.1, [HAuCl4] = 0.20 mM and [AgNO3] = 0.11 mM as the optimal parameters. Then we set as a second criterion that the pH was restricted to the range 7.25-7.35. This was done to find the optimal ratio for the identified optimal growth medium: MH broth (pH = 7.3 ± 0.05). This was the medium that was used in the second iteration for the biological nanoparticle synthesis. The found conditions were: pH 7.25, [HAuCl4] = 0.19 mM, and [AgNO3] = 0.10 mM. With this, the ratio between the gold and silver concentration was 1.85 to 1. For the experiments in the second iteration, we implemented this ratio. The dynamic graph below visualizes the modeling predictions.
Experimental validation
To test the model's predictions, nanoparticles were synthesized with the optimal conditions found in the previous section. Fig. 15 gives the experimental data and the prediction of the model. It is visible that the model overpredicts and that the peaks of the model and the experimental data aren’t at the same wavelength. This could be caused by the model overestimating the influence of certain parameters on the A800 nm.
Fig. 15 | Comparison between the model and experimental data.
Iteration two: Biological nanoparticle synthesis
Determining the experimental parameters and their ranges
The pH range is 7±2.
Assuming that better protein function will result in better nanoparticle formation, the optimal growth pH for E. coli was used as the central value: pH 7 10. The adequate range was further restricted by the biologically relevant pH values for E. coli: growth is only possible between pH 5 and 9 11. Therefore, our tested pH values were 7±2.
The temperature range is 37±12°C
Since 37°C is the optimal temperature for E. coli growth it was chosen as the central value for the temperature range 7. In addition, literature showed that nanoparticle formation benefited from higher temperatures 14. Therefore, the upper boundary was set to 49°C, which was the highest the shakers in our lab could achieve. Finally, according to Box-Behnken Design, an equidistant point was selected, which made the final temperature range: 37±12°C.
Important note
In the second iteration, we used both gold and silver salts in the experiments. However, we only mention the concentration of gold salts as the adjusted variable since the silver concentration can be calculated by dividing the gold concentration by 1.85, which was the ratio obtained in iteration one.
The gold salts concentration range is 4±2 mM.
After obtaining the optimal gold:silver salts ratio in the first iteration (1.85:1), it was necessary to establish which concentration of gold and silver salts was optimal for PTT-optimized nanoparticle production. To identify an adequate concentration range to be tested in the modeling, it was necessary to conduct a preliminary experiment. The preliminarily tested range for the concentration of the golden salt was 0-10 mM. The results showed that A800 nm decreased as the concentration increased (Fig. 17). Therefore, it was decided to test the 4±2 mM range for modeling.
Fig. 16 | The effect of HAuCl4 concentration on the absorption spectrum of nanoparticles. Experiments were performed with E. coli BL21 supernatant and varying concentrations of HAuCl4. The absorption spectrum was measured with a plate-reader in a 24 well plate after 24 hours. All gold concentration experiments, except for 0 mM gold, were performed with three biological replicates. The dotted lines represent the standard error. A: Full absorption spectrum. B: Same measurement, barplot of the A800 nm added for clarity.
Below are the parameter ranges with the central point we used for the modeling experiments of the second iteration.
| Level | |||
| Variable | -1 | 0 | 1 |
| pH | 5 | 7 | 9 |
| Temperature (°C) | 25 | 37 | 49 |
| [HAuCl4] (mM) | 2 | 4 | 6 |
Experimental procedure
In the second iteration, we performed nanoparticle synthesis in three different solutions to compare their outcome. The first solution consisted of gold and silver salts with WT E. coli BL21 supernatant and WT E. coli BL21 lysate. We refer to this solution as it2-lysate. The second solution consisted of gold and silver salts with WT E. coli BL21 supernatant without lysate. This solution we call it2-SN. All E. coli used in these experiments was grown in MH broth. The last solution was a control and it consisted of gold and silver salts in ddH2O. We refer to this solution as it2-ddH2O. Experiments for all three solutions were performed with the same synthesis conditions. These can be found in Table 14. The results of the experiments are in the dropdown.
Solutions used in iteration two for nanoparticle synthesis
It2-lysate: WT E. coli BL21 supernatant + WT E. coli BL21 lysate + gold and silver salts
It2-SN: WT E. coli BL21 supernatant + gold and silver salts
It2-ddH2O: ddH2O + gold and silver salts
| Run | pH | Temperature °C | [HAuCl4] (mM) |
| 1 | 5 | 25 | 4 |
| 2 | 9 | 25 | 4 |
| 3 | 5 | 49 | 4 |
| 4 | 9 | 49 | 4 |
| 5 | 5 | 37 | 2 |
| 6 | 9 | 37 | 2 |
| 7 | 5 | 37 | 6 |
| 8 | 9 | 37 | 6 |
| 9 | 7 | 25 | 2 |
| 10 | 7 | 49 | 2 |
| 11 | 7 | 25 | 6 |
| 12 | 7 | 49 | 6 |
| 13 | 7 | 37 | 4 |
| 14 | 7 | 37 | 4 |
| 15 | 7 | 37 | 4 |
Table 15 | The it2-lysate’s absorbance between 450 and 1100nm for the different synthesis conditions. For the data see google sheets Download Table 15 here
Table 16 | The it2-SN’s absorbance between 450 and 1100nm for the different synthesis conditions. For the data see google sheets Download Table 16 here
Table 17 | The it2-ddH2O’s absorbance between 450 and 1100nm for the different synthesis conditions. For the data see google sheets Download Table 17 here
Data analysis
The same steps were undertaken as in iteration one. Table 18 shows the transformation that we used and the form of the model.
| Solution | Transformation | Model of the form |
| It2-lysate | Logarithmic | \(ln(A_{800 nm}) = c_{lys_1}\textrm{pH} + c_{lys_2} \textrm{[HAuCl}_4] + c_{lys_3}\textrm{Temperature}\) |
| It2-SN | Logarithmic | \(ln(A_{800 nm}) = c_{SN_1}\textrm{pH} + c_{SN_2} \textrm{[HAuCl}_4] + c_{SN_3}\textrm{Temperature}\) |
| It2-ddH2O | No transformation | \(ln(A_{800 nm}) = c_{dd_1}\textrm{pH} + c_{dd_2} \textrm{[HAuCl}_4] + c_{dd_3}\textrm{Temperature}\) |
The p-values of the models are in Table 19. For this iteration too they are all below 0.05. The full explanation of our choices can be found in the dropdown below.
| Solution | p-value |
| It2-lysate | <10-4 |
| It2-SN | 5.5*10-3 |
| It2-ddH2O | 1.74*10-2 |
All the steps undertaken to find a correct model for each solution are explained here.
Analysis of the A800 nm data of the it2-lysate solution.
Fig. 17 | Box-Cox plot for the it2-lysate A800 nm data. The green line represents the best λ (-0.03). The red lines represent the confidence interval (-0.66, 0.59), and the blue line represents the current transformation (1).
For the A800 nm data of the it2-lysate solution, the software recommended a logarithmic transformation (Fig. 17). We compared performing no transformation and the logarithmic transformation. A linear model was recommended for both transformations. We based our choice upon the normal plot for the two transformations (Fig. 18). When there was no transformation applied, one measurement falls far outside the line. However, there is no strong outlier when a logarithmic transformation is applied.
Fig. 18 | The normal probability plot of the it2-lysate A800 nm data. Left: no transformation. Right: logarithmic transformation.
Therefore, we chose to perform a logarithmic transformation and apply a linear model. Table 20 shows the ANOVA table for our model. Since all terms were significant, the model wasn’t adjusted.
Table 20 | ANOVA table of the it2-lysate A800 nm data. No transformation was applied. The equation is linear.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
3.25 |
3 |
1.08 |
23.82 |
0.0001 |
| A-pH |
1.07 |
1 |
1.07 |
23.43 |
0.0005 |
| B-Gold |
1.95 |
1 |
1.95 |
42.82 |
0.0001 |
| C-Temperature |
0.2365 |
1 |
0.2365 |
5.2 |
0.0435 |
| Residual |
0.5004 |
11 |
0.0455 |
Analysis of the A800 nm data of the it2-SN solution.
Fig. 19 | Box-Cox plot for the it2-SN A800 nm data. The green line represents the best λ (0.15). The red lines represent the confidence interval (-1.08, 1.46), and the blue line represents the current transformation (1).
Based on the Box-Cox plot (Fig. 19), the logarithmic transformation was implemented.
Fig. 20 | The normal probability plot of the it2-SN A800 nm data. A logarithmic transformation was applied.
We see in Fig. 20 that the points are distributed linearly. The software recommended using a linear model. Although the p-values of the pH and temperature (Table 21) are above 0.05, we choose not to remove them in order to still account for the influence of pH and temperature on the absorbance.
Table 21 | ANOVA table of the it2-SN A800 nm data. A logarithmic transformation was applied.The equation is linear.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
1.33 |
3 |
0.4434 |
7.41 |
0.0055 |
| A-pH |
0.041 |
1 |
0.041 |
0.6852 |
0.4254 |
| B-Gold |
1.29 |
1 |
1.29 |
21.48 |
0.0007 |
| C-Temperature |
0.004 |
1 |
0.004 |
0.0661 |
0.8018 |
| Residual |
0.6583 |
11 |
0.0598 |
Analysis of the A800 nm data of the it2-ddH2O solution.
Fig. 21 | Box-Cox plot for the it2-ddH2O A800 nm data. The green line represents the best λ (-0.13). The red lines represent the confidence interval (-2.24, 1.98), and the blue line represents the current transformation (1).
Based on the Box-Cox plot (Fig. 21), the software recommended performing a logarithmic transformation. Due to the outliers in the normal plot (Fig. 22) we also considered performing no transformation.
Fig. 22 | The normal probability plot of the it2-ddH2O A800 nm data. No transformation was applied.
However, as is visible in Fig. 23, the outliers are even more evident when no transformation is applied. Therefore, we chose to apply a logarithmic transformation.
Fig. 23 | The normal probability plot of the it2-ddH2O A800 nm data. A logarithmic transformation was applied.
Although the term for temperature has a p-value above 0.05 (Table 22), we chose to keep the term to show the influence of temperature on the absorbance.
Table 22 | ANOVA table of the it2-ddH2O A800 nm data. A logarithmic transformation was applied.
| Source | Sum of Squares | df | Mean Square | F-value | p-value |
| Model |
0.7622 |
3 |
0.2541 |
5.47 |
0.0174 |
| A-pH |
0.2893 |
1 |
0.2893 |
6.23 |
0.0317 |
| B-Gold |
0.3049 |
1 |
0.3049 |
6.57 |
0.0282 |
| C-Temperature |
0.168 |
1 |
0.168 |
3.62 |
0.0863 |
| Residual |
0.4643 |
10 |
0.0464 |
Results
For the second iteration, the only optimization criterion was the maximization of A800 nm. This gave the optimal values of pH 5.2, a HAuCl4 concentration of 5.75 mM, and a temperature of 49°C. The dynamic graph below visualizes the modeling predictions.
Experimental validation
Although we did not have time to perform experiments to validate the model, we did compare our predicted optimum conditions to those in literature. Divya et al. conducted a study in which pH (5, 7, and 9) and temperature were independently varied to optimize silver nanoparticle synthesis with E. coli4 pH 5 was found to be optimal, which agrees with the optimal pH of 5.2 we found. The authors tested 25°C and 37°C, of which 37°C performed better. In our experiments, too, higher temperatures were found to be more optimal.
Conclusion
By feeding the data into BBD, the optimal conditions for nanoparticle formation were defined: pH 5.2, [HAuCl4] = 5.75 mM, a [AgNO3] = 3.11 mM, and a temperature of 49°C. This knowledge elucidates the optimal nanoparticle synthesis conditions.
Contribution to future teams
Outline
In our contribution to future teams, we outline (1) how to operate the software we used, (2) how to choose factors and their ranges, and (3) how to analyze the data. This will allow future teams to get a complete picture of applying BBD to optimize your experiment.
Sources
It is crucial to consult scientific literature and other sources before starting with modeling to better understand what BBD entails. For example, a review paper by Ferreira et al. gives a good overview of the theoretical background of BBD6.
Determining factors
Firstly, it’s essential to determine the factors that influence your production process. Then, consider that the number of experiments needed scales with the number of factors you’re testing. Therefore, try limiting the number of variables. Alternatively, you can distribute the experiments over multiple iterations, as we have done.
Determining the range of the factors
Your experiments must be performed within the optimal range of the parameters to apply BBD. Therefore, determine these ranges individually before starting your BBD experiment. These typically can be found in the available literature. If this is not the case, try to test a wide range of values.
Accessing and using the software
A license of Stat-Ease Design Expert can be requested on statease.com.
Once the software is downloaded, open it and click on New Design.
On the sidebar navigate to Response Surface >> Randomized >> Box-Behnken. Here you can adjust the amount of factors, their names, ranges, units, and the amount of central points.
Clicking Next will bring you to a page where you can adjust information about responses. You can add more responses and fill in their units.
By clicking Finish you will get to the page where you can fill in your responses. If you have a lot of responses you want to add, it can be tedious to add them manually. You can simply copy paste them into the table or add them by clicking file >> Import from file.
Data analysis
Before you create your final model, your data may need to undergo a transformation. Lee et al. provides a good background on what a data transformation is and how to find the correct one15. A transformation allows your data to be normally distributed. First, determine if the data is normally distributed by plotting the data in a histogram and perform a test to see if the variance is constant. If the data is not normally distributed, a transformation is needed in order to perform statistical analysis. Design Expert recommends transformations for a ratio from min to max that is greater than 10. There are two ways to find the correct transformation. Design Expert gives the recommended transformation Go to Diagnostics >> Report, that will state the recommended transformation. A second way is to look at the Box-Cox plot. The software gives the best λ. In our Data Analysis section is a table of which λ corresponds to which transformation.
After your data is transformed go to Fit Summary. The software will recommend which model you should use.
On the ANOVA >> Analysis of Variants page you can find the ANOVA table of your model. The sum of squares gives an indication of the variance of the data from the mean. df stands for degrees of freedom. It is the number of independent pieces of information used in the calculation of a statistic. The mean square is the sum of squares divided by the degrees of freedom. The F-value is the variation between sample means divided by the variance within the sample. Finally, the p-value describes the likelihood of obtaining the data by random chance. If you want to remove a certain term from your model, for example because you deem its p-value to be too high, navigate to the Model page. There you can double click on a term to remove it.
In the Diagnostics section, you can find the normal plot and many more plots that provide information about your data.
Results
Navigate to the Model Graphs section to get response surface plots of your model.
Going to ANOVA >> Actual Equation or Coded equation gives you the equation of your model.
Optimization
You can find the optimal conditions of your experiment by going to Optimization >> Numerical. For each factor and response, you can decide to add criteria to them, for example maximization and minimization. You can also add importance to the criteria, so that your model knows what is most important to optimize.
If you navigate to the Solutions header you can find the solutions the model found. On the left you can find the levels of your factors and on the right the value of your response. The software also gives the desirability of the outcome.
Data Validation
Don’t forget to validate the model after you’ve obtained the optimal conditions.
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