Team:UCL/Model/Dynamic

Introduction In vivo In vitro Comparison

Dynamic Modelling

Introduction

Another important aspect to consider in our project is dynamic modelling. Whilst prediction of the behaviour of a complex expression system can be useful, from a production-based point of view, comparison between in vitro and in vivo systems is of greater interest as it offers information regarding which system is more fit for upscaling. As such, we have developed cellCONOMY as our in vivo model and, with further simplifications and adaptations, it was converted into an in vitro cell-free protein synthesis model.

cellCONOMY, at its basis, is a mathematical model derived from ordinary non-linear differential equations (ODEs) of the molecular species of interest present in chosen reaction schemes (see below). All presented simulations were carried using Python's odeint function in Jupyter Notebook and cellCONOMY's script can be found here.

In vivo cellCONOMY

The in vivo component of cellCONOMY is a whole cell model based on that described by Weiße et al.1 and looks into the cellular trade-offs caused by:

  1. Limited number of ribosomes
  2. Finite proteome (i.e. constant cell mass)
  3. Finite nutrient levels

The biochemical reaction scheme (Fig. 1) can be broken down into 38 processes (Table 1) which follow the dynamics of the following cell variables: cell number (N), internal nutrient (si), external nutrient (s), ATP (a), ribosomes (r), transporting enzymes (et), metabolic enzymes (em), house-keeping/growth independent enzymes (q), heterologous protein (p) their mRNA transcripts (mx) and the corresponding ribosome:mRNA complexes (cx).


Fig. 1- Reaction scheme for the in vivo component of cellCONOMY. Adapted from Weiße et al.1 Transporter enzymes are presented in orange, metabolic enzymes in green, ribosomes in red, house-keeping enzymes in light blue and heterologous protein in gold.


Table 1- Reaction scheme breakdown for Fig. 1. Adapted from Weiße et al.1

The following assumptions were made in building the in vivo component of cellCONOMY:

  1. External nutrient is uptaken in the cell via transporter enzymes and enters the internal nutrient pool.
  2. Reducing power, energy, amino acids and nucleotides are treated altogether as a primary metabolite (a) which is obtained by catalytic conversion of the internal nutrient by metabolic enzymes.
  3. Energy consumption is attributed only to translation of proteins, but transcription rates are modelled as energy level dependent (ωx).
  4. All transcripts compete for binding to the ribosomes, albeit they all are modelled with the same binding/unbinding rates through mass action kinetics (kb=ku). Polysome formation is neglected, hence each ribosome has to individually bind a mRNA transcript mx.
  5. Housekeeping enzymes (q) are under negative autoregulation in order to maintain stable levels regardless of growth phase of the culture.
  6. Mass of the cell (M) is considered constant throughout its growth.
  7. No explicit protein degradation is modelled, only negative terms for the ordinary differential equations originating from first-order dilution by cell growth (λ). First-order degradation of mRNA transcripts is included (dm).
  8. Heterologous protein expression is considered from the early stages of growth.
  9. External nutrient flow into the medium is modelled and is a direct analogy to continuous-flow cultures. If influx and efflux are made 0, the model simulates a batch culture.

    The corresponding ordinary differential equations (ODEs) derived from the reaction scheme are presented below:

    And the following parameters have been used for our model and have been adapted from Weiße et al.1(Table 2):


    Table 2- Parameter values used for cellCONOMY. Adapted from Weiße et al.1

    Application to lab data

    In order to better optimize cellCONOMY, the model was used to fit growth curves obtained in the lab for the production of our drug delivery platform(Fig. 2). Starting from a 50mL batch cell culture with 0.1 OD units, optical density at 600nm was recorded for every 30 min except for two time points (270min and 300min). By converting the OD to cell count (1 OD unit=8*108 cells/mL), the model was run starting with the same effective population level and, all the initial values for the state variables were determined through trial-and-error until the closest fit to the lab data was obtained. It can be seen in Fig. 2 that the simulated model matches the experimental data quite well for the first 330 min. The divergence point comes from the moment when the cell culture starts to decay, whereas the model still continues its slightly attenuated growth trend.


    Fig. 2 Cell growth experimental data and the fitted cellCONOMY output

    Additional refinements can be brought to the model by changing death rates and initial nutrient values but its stability for over 5 h can still be used to predict the behaviour of well-behaved cell cultures. Furthermore, as seen in Fig. 3 encapsulin-iLOV production was modelled for individual cell and for the batch culture and yielded a final concentration of 53.7 mg/L. This is approximately 3-fold lower than what experimental values were obtained (180 mg/L) and could be attributed to different partitioning of cell resources upon induction and a more pronounced funnelling of transcription and translation towards heterologous protein production. More insights into cell metabolism modelling could further optimise cellCONOMY.


    Fig. 3 Encapsulin-iLOV levels as modelled by cellCONOMY for single cell and batch culture

    In conclusion, the in vivo component of cellCONOMY can be considered a suitable model for transformed colonies which have just been induced and inoculated in a fresh batch culture as it provides a fairly accurate prediction for cell growth and an estimation of the protein yield within less than an order of magnitude below experimental values.

    In vitro cellCONOMY

    Cell free expression systems are attractive alternatives to the regular cell cultures as it contains only the components essential to protein production: the transcription and translation machineries. These systems allow fast-tracking of purification of protein of interest as well as tackle the issue of biosafety by not using live bacteria in the process. While in this situation heterologous protein expression is unhindered by other competing processes (as seen in the in vivo whole cell model), this system lacks the ability to dynamically adapt to different environment conditions, maintain or replenish any molecular machines once they have degraded. Having this in mind, we have derived an in vitro cell-free synthesis model from the in vivo whole-cell one to ensure accurate comparisons between the 2 protein expression systems. The following assumptions were made for the in vitro model:

    1. Import of nutrient and its conversion into primary metabolite has been neglected as the system is assumed to be open and have access to the primary metabolite without requiring processing by metabolic enzymes. Hence, all reactions involving external/internal nutrient (s/si), transporter enzymes (et) and metabolic enzymes (em) have been subtracted from the in vivo model.
    2. As the system is cell-free and contains only the transcription and translation machinery as active enzymes, all processes involving house-keeping enzymes (q) were dismissed.
    3. The only DNA that undergoes transcription is that of added plasmid DNA containing the sequence of the protein of interest, hence transcription and translation of new ribosomes (r) is dismissed.
    4. Only the reactions that affect protein production as well as ribosome degradation were used for the in vitro model.
    5. In line with the in vivo component of cellCONOMY, same state variables related to the heterologous protein were used: protein (p), transcripts of the protein (mp) and the corresponding mRNA:ribosome complex (cp) copy numbers.
    6. Due to the lack of protein maintenance systems such as chaperones, degradation rates for both ribosomes and heterologously-expressed protein were set at dp=0.02 min-1.
    7. Primary metabolite will be considered in excess, such that the simulation will halt from the depletion of ribosomes rather than of energy.

    Thus, the following reaction scheme reflects the in vitro component of cellCONOMY (Fig. 4 and Table 3):


    Fig. 4 Reaction scheme for the in vitro component of cellCONOMY


    Table 3 Breakdown of the reaction scheme presented in Fig. 4

    Note: as the amount of primary metabolite (a) is considered in high excess, the following mathematical simplification can be applied for the derived ordinary differential equations: and . Hence, on top of the assumptions stated previously, the following ODEs have been derived for the model:

    The following parameter values were used for the model(Table 4):


    Table 4 Parameter values used for the in vitro component of cellCONOMY

    Application to lab data

    In order to obtain a sensible estimation of the expected yields of in vitro cell free protein synthesis systems(CFPS), the mathematical model was fitted to experimentally obtained lab data for intein-eGFP production. The lab used CFPS had a total volume of 100μL and a presumed concentration of ribosomes of 2μM2. By numerical conversion, this yielded an effective ribosome count of approximately 1014 which was used as input in in vitro cellCONOMY. Also, it can be seen by comparing the parameter tables for the in vivo and in vitro models that the maximum transcription rate for the expressed protein (wp) is different by 5 orders of magnitude. This latter value was obtained through optimisation of the model fit to the experimental data and was done to artificially mimic the higher availability of plasmid DNA in the CFPS versus the batch culture. Around 2.7*1011 DNA molecules were added in the CFPS whereas a rough prediction indicates only about 200 plasmid molecules for the transformed cells3. It can be seen in Fig. 5 that the cellCONOMY provides a good prediction of protein production between the 50 min and 200 min time points.


    Fig. 5 Protein concentration experimental data and the fitted cellCONOMY output

    As data have been collected based on the fluorescence of the construct, deviations seen at the beginning of the simulation could originate from arbitrary background fluorescence. The deviations seen at the end of the simulation could indicate either a higher degradation rate for ribosomes (dp) used in the model or that ribosomes decay on following a higher-order process. Nonetheless, the differences between the model and the lab data amount to less than 10% for the experimentally obtained value, as for the maximum error at the end of the simulation, it reached a protein yield of 8.88 mg/L in comparison to the experimental value of 9.69 mg/L. Further refinements could be brought to the model by considering the life time and efficiency of the RNA polymerases used in the CFPS as well as gaining a better understanding of the kinetics of the in vitro degradation processes.

    To summarize, in vitro cellCONOMY can be deemed as a suitable model for CFPS systems as it provides sensible predictions of the protein yield, with a maximum error of less than 10% below experimental values.

    in vivo vs in vitro

    A direct comparison between the protein yields of the fitted models is provided in Fig. 6. Here, it can be seen that the protein yields for constructs with similar molecular weights, 46kDa for Encapsulin-iLOV and 57.3kDa for intein-eGFP, vary significantly, from 35.5 mg/L for the in vivo system and 8.88 mg/L for the in vitro system.


    Fig. 6 Comparison of the protein yields between previously fitted cellCONOMY simulations. Note: both simulations were run for 240 min to match the end point of the shorter in vitro simulation of cellCONOMY

    Whilst Fig. 6 alone could argue in favour of the in vivo systems for heterologous protein production, further proof can be brought by rerunning the simulation for proteins of the same length (np). In this case, np=600 to match the molecular weight (65kDa) of our final best composite part: Encapsulin-DARPin and the photosensitiser miniSOG. An overlay of the simulations is presented in Fig. 7:


    Fig. 7 Comparison of predicted yields for Encapsulin-DARPin+miniSOG using cellCONOMY in both modes in the absence (left) or presence (right) of the artificially corrected in vivo simulation (see text below).

    cellCONOMY was run with the new np parameter for 240 min to ensure the highest accuracy of the simulations as possible. Furthermore, the in vivo simulation was multiplied by a factor of 3.3, derived from the ratio between protein yields seen in the batch culture and the simulated in vivo value, to artificially induce higher accuracy for protein production. In the left panel of Fig. 7 it can already be seen that for the same amount of time (240 min) and for the same lab conditions employed previously (50mL batch culture and 100μL CFPS), the in vivo system has higher yields, reaching a value of 34.3 mg/L whereas the in vitro system reached only 8.4 mg/L. The corrected cellCONOMY in vivo simulation seen in the right panel of Fig. 7 reached a final yield of 113.2 mg/L.

    In conclusion, the in vivo systems provide higher yields for heterologous protein production and should be considered for upscaling. Furthermore, this model informed us on the choice of expression system, as all subsequent lab-tested constructs were produced and isolated from cell cultures.

    References

    1. Weiße, A., Oyarzún, D., Danos, V. and Swain, P. (2015). Mechanistic links between cellular trade-offs, gene expression, and growth. Proceedings of the National Academy of Sciences, 112(9), pp.E1038-E1047.
    2. Moore, S., MacDonald, J., Wienecke, S., Ishwarbhai, A., Tsipa, A., Aw, R., Kylilis, N., Bell, D., McClymont, D., Jensen, K., Polizzi, K., Biedendieck, R. and Freemont, P. (2018). Rapid acquisition and model-based analysis of cell-free transcription–translation reactions from nonmodel bacteria. Proceedings of the National Academy of Sciences, 115(19), pp.E4340-E4349.
    3. Part:pSB1C3 in the Registry of Standard Biological Parts, Available at: https://parts.igem.org/Part:pSB1C3 [Accessed 17 Oct. 2019].