. Model .

Introduction

Secretion of recombinant protein from engineered bacteria ties to two components: one is the secretion machinery of engineered bacteria, and the other is the signal peptide or signal sequence. The sequence of recombinant protein is obtained by fusing the sequence of protein of interest (POI) with the signal peptide, in most cases, at the N-terminal of POI. The type of signal peptide is accordance with the specific secretion pathway. In prokaryotic cells, for many application systems, there are two common secretion mechanisms of recombinant proteins: Type I secretion system (T1SS) and Type II secretion system (T2SS). More specially, T2SS contains more sub-type mechanisms, including the Sec (or SecB) pathway, SRP (signal recognition particle) pathway, and Tat (twin-arginine translocation) pathway.

This year, XMU-China tested the performance of multiple signal peptides in the continued growth state of cells, resulting in distinct secretion kinetics compared to traditional logarithmic-shaped curves gained in pulse-chase experiments, the exponential-shaped curve indeed (Figure 1A). Some iGEM teams (Stuttgart 2017, IIT_Kanpur 2018) endeavor to build the kinetic landscape for the secretion in continued growth situations with the empirical formula, showing less understanding of the biological insight for the complexity of translocation. Not to mention that those teams failed in the reconstruction of logarithmic secretion kinetics. Now, it is time to develop a new mathematical model to describe and analyze the secretion kinetics under the practical way in real-world engineering.

The original intention of this model was to mathematically describe the relationship between the amount of secreted recombinant protein and time in the natural state of our engineered bacteria and explore the secretion efficiency of different signal peptides that we used, thus choosing the best combination of signal peptide and Anderson promoter to optimize the economic cost of our deinking process. Such a modeling process was of great significance for us in implementing REPARO and part collection.

Derivation

Here, the multi-scale estimation (MUSES) is proposed for the secretion kinetics of signal peptides combined with both macro-level and micro-level models, focusing on an extremely practical situation in which the cell is in a natural growth state to provide guidelines for the design of related synthetic biosystem.

1. Macro-Level Model

An equation system based on modified Logistic and Monod models was employed to model for the quantity of viable, dead, and lysed cells as follows.

$$ X_{T} = X_{V} + X_{D} $$

$$ \frac{dX_{V}}{dt} = \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)X_{V} - k_{d}X_{V} $$

$$ \frac{dX_{D}}{dt} = k_{T}X_{V}\left( {1 - \frac{X_{V} + X_{D}}{X_{T,max}}} \right) - \left( {\frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right) - k_{d}} \right)X_{V} - k_{dXd}X_{D} $$

$$ \frac{dX_{lys}}{dt} = k_{dXd}X_{D} $$

with $t$ the time (h), $X_T$ the total cell density (OD₆₀₀), $X_V$ the viable cell density (OD₆₀₀), $X_D$ the dead cell density (OD₆₀₀), $X_{lys}$ the lysed cell density (OD₆₀₀), $\mu_{max}$ the maximum specific growth rate (1/h), $K_S$ the saturation constant (1/h), $X_{v,max}$ the viable cell carrying capacity (OD₆₀₀), $k_d$ the specific death rate of viable cells which have no contribution to product (1/h), $k_T$ the intrinsic growth rate in logistic equation (1/h), $X_{T,max}$ the total cell carrying capacity (OD₆₀₀), $k_{dXd}$ the specific lysis rate of dead cells (1/h). $S$ is the concentration of growth-limiting substrate (g/L), which is the nitrogen source (tryptone) in our work since the task is related to protein, the growth-associated product. Thus, the following two equations was written to describe both substrate and product concentration:

$$ \frac{dS}{dt} = \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)\frac{1}{Y_{Xv/s}}X_{V} - k_{ds}S $$

$$ \frac{dP}{dt} = \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)\frac{1}{Y_{P/Xv}}X_{V} - k_{dp}P $$

with $S$ the concentration of substrates (g/L), $P$ the concentration of products (a. u.), $Y_{Xv/s}$ the maximum biomass and product yield based on growth-limiting substrate (g/L/OD₆₀₀), $k_{ds}$ the specific degradation and dilution rate of substrate (1/h), $Y_{P/Xv}$ the maximum product yield based on substrate (g/L/OD₆₀₀), $k_{ds}$ the specific degradation and dilution rate of product (1/h).

2. Micro-Level Model

3.1 For Sec Pathway

For Sec pathway (left in Fig. 3), a soluble molecular chaperone protein named SecB binds to the recombinant protein's signal peptide or other features of its incompletely folded structure. The bound protein is then delivered to SecA, a protein acts as both a receptor and a translocating ATPase that is associated with the cytoplasm membrane. The Sec-recombinant protein complex forms a translocation complex with SecY/SecE/SecG (SecYEG). The protein is conducted by the energy released from ATP hydrolysis and PMF (proton motive force). The SecDF complex (YajC is an accessory protein) plays the role as the signal peptidase (SPase) to cleave the signal peptide in the final stage of translocation. Then the protein maturation will happen in the periplasm space finally. Thus, the micro-level model for Sec pathway-related signal peptide will be written based on mass action law and macro-level model as follows:

$$ \frac{dRNA}{dt} = k_{1} - k_{d1}RNA $$

$$ \frac{dP_{c}}{dt} = k_{2}RNA - k_{d2}P_{c} - \alpha P_{c} $$

$$ \frac{dP_{p}}{dt} = \alpha P_{c} - k_{d3}P_{p} - \beta P_{p} $$

$$ \frac{dP_{l}}{dt} = \beta P_{p} - k_{d4}P_{l} - \gamma P_{l} - \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)\frac{X_{V}}{X_{V} + X_{D}}P_{l} - k_{dXd}\frac{X_{D}}{X_{V} + X_{D}}P_{l} $$

$$ \frac{dP_{M}}{dt} = \gamma P_{l}X_{V}d_{1} + k_{dXd}X_{D}P_{l}d_{2} + \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)X_{V}P_{l}d_{3} $$

with $RNA$ the intensity of target protein's mRNA (g/L/OD₆₀₀), $P_c$ the intensity of unmature protein in cytoplasm space (a. u./OD₆₀₀), $P_p$ the intensity of unmature protein in periplasm space (a. u./OD₆₀₀), $P_l$ the intensity of mature protein in periplasm space (a. u./OD₆₀₀), $P_M$ the concentration of mature protein in medium space (a. u. ), $k_1$ the specific translation rate (g/L/OD₆₀₀/h), $k_{d1}$ the specific degradation and dilution rate of RNA (1/h), $k_2$ the specific transcription rate (g/L/OD₆₀₀/h), $k_{d2}$ the specific degradation and dilution rate of $P_c$ (1/h), $\alpha$ the translocation rate from cytoplasm space to periplasm space (1/h), $k_{d3}$ the specific degradation and dilution rate of $P_p$ (1/h), $\beta$ the protein maturation rate (1/h), $k_{d4}$ the specific degradation and dilution rate of $P_l$ (1/h), $\gamma$ the translocation rate from periplasm space to medium space (1/h), $d_i\ (i=1,2,3,4\cdots)$ the dilution rate due to the volume change between different space (1/h).

3.2 For Tat Pathway

The Tat pathway is commonly used for those proteins that contain two consecutive and highly conserved arginine residues in their leader peptides. On the Tat pathway (right in Fig. 3), the protein is fully synthesized and folds in the cytoplasm where it can bind specific cofactors. The signal peptide is then recognized by TatC in the TatBC complex. Signal peptide binding promotes association of the complex with TatA oligomers at the expense of PMF. Protein translocation occurs through a channel formed by TatA and possibly TatE oligomers. Thus, the micro-level model for Tat pathway-related signal peptide will be written based on mass action law and macro-level model as follows:

$$ \frac{dRNA}{dt} = k_{1} - k_{d1}RNA $$

$$ \frac{dP_{c}}{dt} = k_{2}RNA - k_{d2}P_{c} - \beta P_{c} $$

$$ \frac{dP_{l}}{dt} = \beta P_{c} - k_{d4}P_{l} - \alpha P_{l} - k_{dXd} \frac {X_{D}}{X_{V} + X_{D}} P_{l} $$

$$ \frac{dP_{p}}{dt} = \alpha P_{c} - k_{d3}P_{p} - \gamma P_{p} - \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)\frac{X_{V}}{X_{V} + X_{D}}P_{p} - k_{dXd}\frac{X_{D}}{X_{V} + X_{D}}P_{p} $$

$$ \frac{dP_{M}}{dt} = \gamma P_{p}X_{V}d_{1} + k_{dXd}X_{D}P_{l}d_{2} + \frac{\mu_{max}S}{K_{S} + S}\left( {1 - \frac{X_{V}}{X_{v,max}}} \right)X_{V}P_{p}d_{3} + k_{dXd}X_{D}P_{p}d_{4} $$

with RNA the intensity of target protein's mRNA (g/L/OD₆₀₀), $P_c$ the intensity of unmature protein in cytoplasm space (a. u./OD₆₀₀), $P_l$ the intensity of mature protein in cytoplasm space (a. u./OD₆₀₀), $P_p$ the intensity of mature protein in periplasm space (a. u./OD₆₀₀), $P_M$ the concentration of mature protein in medium space (a. u. ), $k_1$ the specific translation rate (g/L/OD₆₀₀/h), $k_{d1}$ the specific degradation and dilution rate of $RNA$ (1/h), $k_2$ the specific transcription rate (g/L/OD₆₀₀/h), $k_{d2}$ the specific degradation and dilution rate of $P_c$ (1/h), $\alpha$ the translocation rate from cytoplasm space to periplasm space (1/h), $k_{d3}$ the specific degradation and dilution rate of $P_p$ (1/h), $\beta$ the protein maturation rate (1/h), $k_{d4}$ the specific degradation and dilution rate of $P_l$ (1/h), $\gamma$ the translocation rate from periplasm space to medium space (1/h), $d_i\ (i=1,2,3,4\cdots)$ the dilution rate due to the volume change between different space (1/h).

Implementation

Highest Secretion Efficiency and Proper Metabolic Burden of LMT Signal Peptide

The secretion kinetics of various signal peptides from both the Sec pathway (LMT, PelB, OmpA, OsmY, YebF, and AIgen) and the Tat pathway (TorA) were analyzed (Figure 4). The high average R² values for each signal peptide (LMT: 0.994, PelB: 0.993, OmpA: 0.987, OsmY: 0.933, YebF: 0.989, TorA: 0.975, AIgen: 0.988) confirm the accuracy of the multi-scale model in capturing secretion kinetics and demonstrate the successful application of the MUSES method. The experimental results aligned well with the model's predictions (Figure 4A-G), indicating that the multi-scale estimation method effectively deconstructs the dynamic secretion process of signal peptides.

Thus LMT, the best signal peptide among all test groups, whose high secretion efficiency and low metabolic burden are confirmed with our multi-scale model, was chosen to test coupling with our deinking enzymes (the best one in the second screen cycle, CYP199A4 T253E). The SDS-PAGE proves the successful secretion of CYP199A4 T253E (Figure 6A), and the deinking characterization result achieved 64% performance of the high-concentration enzyme sampling prepared by ÄKTA/AKTA (Figure 6B), proving the functionality of our combination. Now, this basic part (BBa_K5136047) designed by combining the high throughput screening and multi-scale signal peptides secretion kinetics, is a crucial part of our part collection.

LMT-mediated Secretion under Constitutive Expression for Optimal Economic Cost

Previous LMT-mediated secretion work in our project is based on T7 promoter, requiring the addition of IPTG to induce the expression of the fusion protein of our deinking enzyme and LMT signal peptide. The IPTG has a toxic effect on cells, which is also expensive in industrial-scale applications. Thus, designing the LMT-mediated secretion system under constitutive expression is essential for the implementation of REPARO by eliminating the extra cost (inducer for example) to achieve optimal economic cost.

Thus, some exapmle in Anderson promoter was selected and tested (Figure 8).

By measuring the fluorescence intensity in the supernatant of each circuit, BBa_K5136202 with promoter J23104 has the most appropriate secretion efficiency, which has enough proteins storage in the intracellular that can be secrete into extracellular environment quickly with the help of secretion, leakiness, and lysis.

Discussion

For our project, the MUSES mathematically describe the relationship between the amount of secreted recombinant protein and time in the natural state of our engineered bacteria and explore the secretion efficiency of different signal peptides that we used, thus choosing the best combination of signal peptide and Anderson promoter to optimize the economic cost of our deinking process.

References

1. M.N.M. Fuad, Applying Logistic and Monod models in a single equations system framework for cell culture growth modeling and estimation, (2020). https://doi.org/10.22541/au.160647305.55974012/v1.
2. R. Freudl, Signal peptides for recombinant protein secretion in bacterial expression systems, Microb. Cell Factories 17 (2018) 52. https://doi.org/10.1186/s12934-018-0901-3.