Summary
Filters are widely used in engineering to reduce noise and/or the magnitude of a signal of interest. Feedback filters, or adaptive filters, are preferred if the signal noise distribution is unknown. One of the main challenges in Synthetic Biology remains the design of reliable constructs but these often fail to work as intended due, e.g. to their inherent stochasticity and burden on the host. Here we design, implement and test experimentally a biological feedback filter module based on small non-coding RNAs (sRNAs) and self-cleaving ribozymes. Mathematical modelling demonstrates that it attenuates noise for a large range of parameters due to negative feedback introduced by the use of ribozymes and sRNA. Our module modifies the steady-state response of the filtered signal, and hence can be used for tuning the feedback strength while also reducing noise. We demonstrated these properties theoretically on the TetR autorepressor, enhanced with our sRNA module.
1 Introduction
Synthetic Biology aims to design new or re-design existing biological devices and systems for a particular purpose. Examples include the design of ‘cellular factories’ producing valuable chemical compounds, biosensors capable of detecting toxins or viruses in a cell culture [Brophy and Voigt, 2014, Purnick and Weiss, 2009, Freemont and Kitney, 2015], or drug delivery systems [Zhou, 2016, Ozdemir et al., 2018]. Exploiting the intracellular machinery allows the synthesis of organic compounds that cannot be easily produced by other means, leading to novel applications in biotechnology, bioprocess engineering and cell-based medicine. However, one of the main challenges in Synthetic Biology remains the design of genetic systems that can be implemented in a predictable and robust way. Due to uncertainty, noise, burden and cross-talk inherent to biological systems, synthetic circuits can fail to work as intended. Indeed, elevated levels of protein production induce a high burden on the cell, notably by sequestering resources for transcription and translation (e.g. RNA polymerases and ribosomes) [Ceroni et al., 2015]. Operating at elevated protein production levels can also increase variability in the protein production due to intrinsic noise. To avoid these issues, common strategies to reduce the level of protein expression are to reduce the strength of promoters, the efficiency of the ribosome-binding site (RBS) or the plasmid copy number. However, transcriptional control is generally system dependent, diminishing the reliability of these approaches.
Filtering techniques are often used in signal processing, feedback control theory and communication systems to reduce signal noise [Haykin, 2002]. Filters can be classified into feedforward and feedback (or adaptive) filters. Feedforward filters are generally used when the noise statistics are known or can be estimated a priori; their output is the difference between the signal of interest and a modification of the same signal. Feedback filters automatically adjust their behaviour by comparing the output signal to the signal of interest at the input of the filter and thus are more favourable for signals corrupted by unknown noise distributions. In the context of Systems and Synthetic Biology, filtering capabilities of signalling cascades [Hooshangi et al., 2005, Thattai and van Oudenaarden, 2002], annihilation motifs [Laurenti et al., 2018] and other motifs [Samoilov et al., 2002] were studied in silico. Feedforward band-pass filters, which pass the signal only in a specific band of frequencies, have been constructed in vivo [Sohka et al., 2009], [Muranaka and Yokobayashi, 2010], while a noise attenuating feedforward filter was proposed and implemented in vitro in [Zechner et al., 2016].
The design of feedback filters is often performed with the help of feedback control theory, which has proven useful to render uncertain systems more reliable and robust to perturbations [Åström and Murray, 2008, Del Vecchio and Murray, 2015, Iglesias and Ingalls, 2010]. In a feedback loop, the output signal is measured and then used to modify the input of the system. In the filter design case, the controlled system is trivial: the signal corrupted by noise. Feedback control theory methods have been successfully applied in synthetic biology previously [Steel et al., 2017b], [Hsiao et al., 2018], [Ang and McMillen, 2013], [Briat et al., 2016], [El-Samad et al., 2002], [Del Vecchio et al., 2008], [Lillacci et al., 2018], [Cantone et al., 2009].
For example, in order to achieve a desired protein expression level an external computer was used to decide the input to the system (chemical or light induction) based on output measurements [Menolascina et al., 2011, Milias-Argeitis et al., 2011, Uhlendorf et al., 2012]. Such systems have inherent drawbacks, as control is achieved by interfacing the living cells with a digital computer that implements the control system.
Over the past few years, focus has shifted towards designing self-contained in vivo controllers. While the vast majority of these experimental implementations were protein-based [Hsiao et al., 2014, Folliard et al., 2017, Rosenfeld et al., 2002] small non-coding RNAs (sRNAs) have also been recently used in this context [Ghodasara and Voigt, 2017, Takahashi et al., 2014, Hu et al., 2018, Kelly et al., 2018]. sRNAs are found in all domains of life and have been shown to play critical regulatory roles in many processes [Cech and Steitz, 2014], [Michaux et al., 2014], [Robledo et al., 2018], [Gottesman and Storz, 2011], [Livny and Waldor, 2007], [Nitzan et al., 2017]. Most sRNAs characterised to date act as post-transcriptional regulators by interacting with specific mRNA targets. Feedback loops involving sRNAs can be found in natural biological processes, for example in the regulation of the expression of quorum-sensing genes [Liu et al., 2013] and in the promotion of a switch for adequate Lrp-dependent adaptation to nutrient availability [Holmqvist et al., 2012]. Post-transcriptional down-regulation is favourable since no proteins are being expressed in this regulation mechanism. Instead, sRNAs are produced quickly, potentially propagating signals rapidly [Holmqvist et al., 2012, Hussein and Lim, 2012, Mehta et al., 2008, Takahashi et al., 2014] and require less energy than proteins, hence reducing the burden to the host. Their operational dynamics are also much faster due to their naturally high degradation rate [Hussein and Lim, 2012]. Therefore, sRNAs provide a promising alternative to the commonly used transcriptional control [Steel et al., 2017, Agrawal et al., 2018].
In this work, we considered two sRNA-based designs to filter variations in transcription, shown in Figure 1. In the first design the regulatory sRNA is placed under the control of a separate promoter to the one controlling transcription of a target gene (henceforth in trans design). In the second design, the sRNA is placed directly downstream of the target gene in cis so that both are under the control of the same inducible promoter. The in cis design also contains a self-cleaving ribozyme between the regulated mRNA and the sRNA sequences, as experiments demonstrated that the mRNA-sRNA strand needs to be separated for the translational attenuation to be efficient. We computationally showed the benefits of the in cis in comparison to the in trans design. While modelling the two circuits and performing numerical simulations showed that the mean steady-state values in both design are attenuated at similar levels, it was evident that the in cis design reduces noise significantly, while the in trans design can adversely amplify it. Modelling also revealed that the in trans design operates approximately as a feedforward filter, in that its output is the mRNA available after sRNA regulation while the in cis design also contains a feedback component, in that the free sRNA produced by self-cleavage of the ribozyme can regulate the amount of mRNA-sRNA transcript available for cleavage.
As the in cis design also attenuates the mean steady-state of the signal, this module can also be used in feedback control in order to reduce the strength of the feedback. We demonstrate the value of the in cis design on the Ptet/TetR autorepressor. Here, sRNA is used to tune the TetR feedback strength without modifying the rest of the system. Our numerical simulations suggest that the in cis design offers a tunable response in terms of the mean output while attenuating transcription noise.
2 Results
2.1 Conceptual designs of sRNA-based filters
We first considered the conceptual designs of the in trans and in cis filters depicted in Figure 1.I (and as block diagrams in Figure 1.II), which can be modelled using a similar set of reactions. In the in trans design we assumed the following reactions: where Prot denotes a protein, which is the filter output. In this design, mRNA and sRNA are transcribed in two different chemical reactions with rates βm, βs, respectively.
In the in cis design, however, mRNA and sRNA are transcribed in the same reaction with the same transcription rate βms, so that this model takes the form
This assumes that the transcribed strand containing mRNA and sRNA splits into mRNA and sRNA instantaneously. This conceptual (or ideal) representation drove the biological implementation discussed later in the text.
In both designs we assumed that mRNA is translated into a protein at the rate kt and that the degradation/dilution rate for every species is different, as mRNA generally degrades faster than proteins and the reported values for the degradation rate of sRNA vary [Hussein and Lim, 2012]. We also assumed that the rate of mRNA-sRNA unbinding is negligibly small, as previously reported [Hussein and Lim, 2012, Kelly et al., 2018], and therefore we did not include it in our model. Modelling both designs using mass-action kinetics yielded the model presented in Figure 1.III, with the difference that for the in cis design, we have βm = βs = βms.
Note that while sRNA down-regulates the translation process in both designs, the two designs lead to different responses to disturbances in the mRNA transcription process. Indeed, the in cis design should be able to attenuate the transcription disturbance better since for every molecule of mRNA produced, so is one molecule of sRNA. Therefore, a burst in transcription of mRNA would also result in a burst in transcription of sRNA. To illustrate the response to disturbances in transcription, we varied the production rate of mRNA in both systems simultaneously, that is we used βm = βms = u(t) = 1 + w(t)[nM/min], where w(t) is the disturbance signal (see Figure 1.IV for the used signals w(t)), and we set βs = 1 [nM/min], krep = 0.5 [1/(nM min)],δm = 0.2476 [1/min], δS = 0.0482 [1/min], δp = 0.0234 [1/min] and kt = 1 [1/min] (see Table S1 in SI). The results of the simulation are shown in Figure 1.IV, where the protein concentrations with βm =, γms = u(t) were divided by the steady-state protein concentrations with βm =, γms = 1 giving the normalised response. The results clearly indicate that the in cis design attenuates the disturbance better than the in trans design.
2.2 Biological implementation of the in cis design
2.2.1 Importance of mRNA-sRNA cleavage in the in cis design
Next we constructed the in cis design in the laboratory and to test experimentally whether controlled attenuation could be achieved using this design. To further minimize the burden on the cell, we chose to use a low copy number plasmid as the vector to implement our in cis RNA-based attenuator design (Table S4 in SI). We also chose to use Ptet as the inducible promoter as it offers tight regulation in response to aTc. As a proof-of-principle, we chose sfGFP as the output to be attenuated. The synthetic regulatory sRNA was designed following the protocol described by [Na et al., 2013, Yoo et al., 2013], in which we changed the binding sequence to target sfgfp. The sequence of our construct hence consists of an sfgfp, ribozyme, the synthetic sRNA consisting of the target binding sequence (TBS) followed by an Hfq-recruiting micC scaffold. Based on [Yoo et al., 2013], we chose a 25-nucleotide long sequence as a starting point for the TBS. Using the web-based service DINAMelt, this sequence gave a ΔG = −30.4 kcal • mol−1, in line with full translation inhibition in [Yoo et al., 2013]. We also hypothesized that the sRNA should be cleaved off the mRNA strand for efficient binding and translation inhibition. We therefore introduced a self-cleaving ribozyme, the Human Hammerhead Ribozyme 9 (HHR9) shown to work well in vivo [De la Peña et al., 2003, De La Peña and García-Robles, 2010, Perreault et al., 2011], between sfgfp and the sRNA.
We monitored cell fluorescence over time in response to varying levels of aTc for two constructs, one with no ribozyme and one carrying HHR9, and compared them with the fluorescence from cells lacking the ribozyme/sRNA part.Figure 2.II shows the steady-state levels of normalized fluorescence for each strain. Attenuation of the output is only observed for the construct expressing the HHR9 ribozyme, confirming our hypothesis that cleavage of the sRNA from the target mRNA is necessary for efficient translation inhibition.
2.2.2 Fine tuning the steady-state level
We next tested the possibility of fine tuning the level of attenuation by modifying the TBS of the sRNA, following the protocol described in [Yoo et al., 2013]. To do so, we decided to either increase or decrease the length of the TBS in the construct with the HHR9 ribozyme, leading to an increase and decrease of the translation inhibition, respectively. We estimated the different binding energies using DINAMelt and chose four different new sequence lengths to test: 30–, 27– and 22–nucleotides long, giving binding energies ΔG = –38.2 kcal • mol−1, ΔG = –31.6 kcal ·mol−1, ΔG = –28.6 kcal ·mol−1, respectively. We monitored the cell fluorescence over time in response to varying levels of aTc for each construct.Figure 2.III shows the output of the system for the different binding energies, displayed as the normalised fluorescence plotted against different aTc concentrations. The output can be reduced to 40% of the signal (for the longest TBS tested) and its value can be varied by altering the length of the TBS, as predicted.
2.3 Modelling and analysis of the in cis filter
2.3.1 Modelling the in cis filter
Having established that the conceptual designs can be implemented experimentally, we proceeded with a more detailed mathematical model to understand further their properties. For convenience we labelled the mRNA of GFP as mGFP. We assumed that self-cleavage of the ribozyme takes place after transcription of the full RNA, that is, mGFP-ribozyme-sRNA (labelled fmRNA) is cleaved into mGFP and sRNA with a rate krc. We assumed that sRNA binds to mGFP preventing GFP translation. We also assumed that sRNA can bind to fmRNA, which can then self-cleave into sRNA and an mGFP-sRNA complex. Since experimental data suggests that the presence of a ribozyme is essential for sRNA and mRNA binding in the in cis design, we assumed that fmRNA (mGFP-ribozyme-sRNA strand) does not bind to the target mRNA (mGFP). We assumed that GFP can be translated both from mGFP and fmRNA. The other reactions were assumed to be the same as for the in trans design, leading to the following chemical reaction model:
We followed the standard mass-action kinetics modelling framework and obtained the model presented in Figure 3.III. We analysed the resulting model as described in the SI. In particular, the frequency domain analysis showed that both in cis and in trans designs implement a low-pass filter attenuating high frequency noise. For realistic parameter values the term krc c — krep sf in Figure 3 remains close to zero, therefore, it does not significantly affect the equation of the sRNA concentration (s) and we hence pictorially represent that sRNA directly degrades fmRNA in Figure 3.I. Depicting the in cis design in the block diagram in Figure 3.II revealed the structure of the filter. The mGFP and sRNA interaction represents the feedforward part of the filter from the transcription initiation, since sRNA and mGFP are produced at similar time instances and sRNA binds to mRNA forming an inert complex. There is also a feedback part in this design formed by the sRNA and the fmRNA interaction. Indeed, fmRNA self-cleaves into mGFP and sRNA, which then binds to fmRNA forming the complex, which contains a ribozyme and splits to an inert complex mGFP-sRNA and a free sRNA.
We also derived a non-dimensional model of the in cis design, which clearly exhibited timescale separation between the quantities f + m, s, p on one side and f, c on the other (see SI for details). This allowed the derivation of a simplified deterministic model of the in cis filter where mtot is the total concentration of fmRNA and mGFP. The key assumption for this analysis was the faster ribozyme cleavage rate in comparison to other reactions. Further investigation revealed strong stability properties of the simplified model, in particular, we ruled out oscillations and multiple steady-states under some assumptions.
Our analysis suggests a possible tuning dial in the in cis design: the ribozyme (with cleavage rate krc) can be used to adjust the gain of the output attenuation, as well as the sRNA-mGFP binding strength krep. With the ribozyme cleavage rate increasing, the deterministic model for this system converges to the ‘ideal’ model of the conceptual design, however, the reported values of the ribozyme cleavage rate krc are not large enough for us to assume that and so that we cannot discard the ribozyme cleavage rate completely. While the simplified model was useful for the analysis and revealed the mathematical difference between the in cis and in trans designs, it hid the feedback part of the filter. This raised the question if the feedback part of the filter has a significant effect on the repression of translation.
2.3.2 In silico evidence of the feedback in the in cis design
Here we evaluated the influence of the feedback on the repression of translation. We performed model simulations of the in cis design, and the models of in cis design without the feedforward part (mGFP and sRNA binding) and without the feedback part (fmRNA and sRNA binding). We set krep = 0.5 [1/(nM min)], krc = 5 [1/min] δm = 0.2476 [1/min], δs = 0.048 [1/min], δp = 0.0234 [1/min] and kt = 1 [1/min] (see Table S1 in SI). We replaced the production rate of fmRNA βms in all three systems with , where u(t) is the disturbance signal depicted by dashed purple line in Figure 4.I and = [0.1, 0.5, 1] [nM/min]. We plot the response of the systems divided by the response with u(t) = 1. Numerical simulations presented in Figure 4.I clearly suggest that the feedback part of the filter has a larger influence on the steady-state behaviour than the feedforward part even with a high ribozyme cleavage rate krc = 5 [1/min].
2.3.3 In cis filter improves the noise properties of the signal
The simulations of the conceptual model suggest that the in cis design attenuates intrinsic noise of the promoter in a much more efficient way than the in trans design. We verified this hypothesis by performing stochastic simulations using the Gillespie Algorithm with the parameters/parameter ranges in Table S1. We considered the coefficient of variation as a noise metric (Figure 5). We plotted the coefficient of variation relative to the mean steady-state for each design. These numerical simulations suggest that the in trans design has a very narrow range of krep values for which noise is attenuated, when compared to the circuit with no sRNA (or krep = 0) while the in cis design attenuates noise for almost all values of krep. An example is presented in the caption of Figure 5, while the numerical values are given in Table S2 in SI. This analysis suggests a simple method to design the in cis filter: choose the maximum possible combination of βms, krep that achieves the desired GFP mean values.
Additional simulations (see Figure S3 in SI) for the in cis design revealed that the level of noise attenuation can be tuned by several parameters: the repression strength krep, the ribozyme cleavage rate krc and the degradation rate of sRNA δs. In particular, increasing the ribozyme cleavage rate krc or the sRNA degradation rate δs lead to a decrease in the noise levels.
2.4 In cis module tunes the feedback strength and reduces noise in the TetR autorepressor
We then proceeded to investigate how the two modules behave in a feedback interconnection, such as for example when an tetR-gfp fusion gene is placed under the control of a Ptet promoter. In this case, we expect the TetR being produced to repress the activity of Ptet (Figure 6.I). We consider the following chemical reactions for the in trans design:
Here, GFP production is not modelled since it is fused to TetR and only serves as a reporter on TetR production. Both TetR and sRNA are controlled by Ptet (see SI for a full model description). We assume that the rest of the interactions follow mass action kinetics.
We assume that βt = βs to aid comparison with the in cis design, which can be modelled using the following chemical reactions:
For the stochastic simulations in Figure 6.II we used the parameters/parameter ranges in Table S1 and additionally set krc = 1 [1/min], kt = 1 [1/min]. These simulations suggest that the in cis design attenuates noise better in comparison with the no sRNA (classical autorepressor) circuit for a wider range of parameters than the in trans design. Note that with sufficient increase of the feedback strength the noise levels can be amplified by the in trans design, which is consistent with previous studies [Kelly et al., 2018]. In our in cis design the noise amplification does not occur for the simulated range of parameters (noise amplification is still possible for larger aTc concentrations). Furthermore, for a particular mean TetR level we can always select a combination of the sRNA repression strength krep and the aTc concentration so that the coefficient of variation is reduced in comparison to ‘no sRNA’ (see caption toFigure 6.II). In the in trans design these tuning dials are less effective: the noise reduction can be insignificant or the sRNA repression strength is very small, which means that the in trans approach is not appropriate for noise reduction. The noise analysis suggests that the repression rate krep adds a valuable tuning dial to the feedback strength design along with the aTc concentration. The numerical values of these simulations are given in Table S3 in SI.
3 Discussion
In this paper, we report the design of an sRNA-based feedback filter where the regulatory sRNA is placed directly after the gene to regulate in cis the signal, resulting in a filtered output. Modelling this new design, we showed that it can improve noise attenuation significantly compared to an in trans filter design and a no filter (no sRNA) design. Our results clearly indicate that the in cis design adapts better to the inputs than the in trans design mainly due to the presence of the feedback component. Moreover, in the in cis system, the production rate of the mRNA and sRNA change simultaneously, attenuating the transcription disturbance better than in the in trans design, where the relative gene expression rate varies significantly due to the sRNA and mRNA transcription rate being decoupled. Lastly, our in cis design requires less cellular resources (e.g. RNA polymerase), decreasing the burden imposed on the cell.
We successfully implemented this new sRNA-based filter in vivo. Our approach, using synthetic sRNA as described by [Na et al., 2013, Yoo et al., 2013] allows not only attenuation but also fine tuning to a desired output. Indeed, altering the length of the target binding sequence (TBS) allows varying the strength of the sRNA-mRNA binding, therefore leading to different levels of attenuation. We tested several length (from 22 to 30 nucleotides long) and could attenuate the output of the filter down to 40% of the unregulated output, very close to the values reported in other in trans designs [Kelly et al., 2018]. Increasing the length of the TBS should in theory allow higher attenuation levels, although off-target binding might then have to be taken in account [Na et al., 2013, Yoo et al., 2013]. Recently a similar architecture was proposed in mammalian cells [Lillacci et al., 2018], where micro RNA was placed in cis with the regulated gene. In our system, placing the syntethic sRNA in cis with the target mRNA without the ribozyme did not yield positive results. We showed, however, that the targeted mRNA and the regulatory sRNA have to be cleaved from each other for efficient output attenuation. Such cleavage was achieved by placing a self-cleaving hammerhead ribozyme (the HHR9 ribozyme) between the mRNA and the sRNA. The ribozyme represents another tuning dial allowing further fine tuning of the system. The ribozyme/synthetic sRNA approach, other than providing tuning dials such as the ribozyme cleavage rate and the repression strength, has another advantage: the repressing molecule is free from the active one, limiting possible unwanted effects. The emergence of synthetic ribozymes (self-cleaving or cleaving in response to a signal) should allow greater tuning flexibility.
Modelling both the in cis and in trans designs showed the clear advantages of the former design over the latter. While the mean steady-state behaviour of the two designs is quantitatively similar, the noise levels differ. In particular, the in cis design attenuates the transcription noise more efficiently thanks to the simultaneous bursts in transcription for the sRNA and the mRNA and the presence of feedback. Modelling suggests that the feedback strength in the filter is proportional to the ribozyme cleavage rate adding another benefit to the development of synthetic fast-cleaving ribozymes.
Further theoretical analysis showed that our design is a useful tool for feedback control design. We showed that the in cis design is well suited to tune down the feedback strength in a transcriptional based controller such as the TetR autorepressor. Again, the in cis design has superior noise properties in comparison to the in trans design. These findings are consistent with previously reported studies [Laurenti et al., 2018], where a Linear Noise Approximation [Van Kampen, 2007] was used to perform the noise attenuation analysis. Indeed, in a feedback setting, a given mean steady-state value can be achieved through either acting on the signal level (in our case aTc) or the strength of the feedback (in our case mRNA-sRNA binding): the in cis design offers a wide range of parameters achieving the same mean steady-state values with lower noise levels.
In this paper we presented a new sRNA-based feedback filter module. Together with the fast dynamics at which RNA operates, our in cis architecture is a simple, modular and tunable construct that can be applied in a wide range of synthetic biology applications while keeping the burden imposed on the cell at a minimum level.
Author Contributions
ND and AS contributed equally to this work. Conceptualization, GHW and AP; Methodology, ND, AS, AP, GHW; Investigation, ND; Formal Analysis, AS; Writing – Original Draft, ND, AS; Writing – Review & Editing, ND, AS, AP, GHW; Funding Acquisition, AP; Resources, AP and GHW.
Declaration of Interests
The authors declare no competing interests.
4 Star Methods
4.1 Key resources table
4.2 Contact for resource sharing
Further information and requests for resources should be addressed to Prof Papachristodoulou ANTONIS{at}ENG.OX.AC.UK.
4.3 Method Details
4.3.1 Bacterial strains and plasmids
Escherichia coli MG1655 cells were used throughout this entire study unless stated otherwise. Plasmids were produced using standard cloning techniques. All synthetic DNA fragments (gBlocks) and primers used in this study were synthesized by Integrated DNA Technologies Inc. The length of the target binding sequences within the sRNA sequence were estimated using the web-based service DINAMelt (http://mfold.rna.albany.edu/?q=DINAMelt/Two-state-melting). We used pBbS2a-RFP (JBEI-2549, shared by Prof. J. Keasling) as a backbone for all the plasmids made for this work [Lee et al., 2011]. A list and a description of plasmids used in this study can be found in Table S4 in SI. Sequences of all plasmids have been submitted to GenBank. Full details are provided in SI.
4.3.2 Growth conditions and assays
Cells were grown overnight from single colonies to stationary phase in minimal medium 9 (M9) complemented with thiamine 0.34 mg/mL and ampicillin (100μg/mL) at 30° C with shaking and then diluted 1/100 into fresh M9 with ampicillin (100μg/mL) of cells were then loaded onto a 96-well plate (Corning) and left to grow for 2h at 30° C with shaking in a FLUOstar Omega Microplate Reader (BMG LABTECH). After this time, anhydrous tetracycline (aTc) at the appropriate concentration was added to the cells and measurements were acquired in the plate reader (gain: 1000). Absorbance and GFP fluorescence (excitation and emission wavelengths: 485 and 530 nm, with20 nm bandwidth, respectively) were measured every 3 minutes. Fluorescence was normalised by absorbance and plotted over time.
4.3.3 Mathematical modelling
We used mass action and Hill kinetic formalisms in order to model the chemical reactions as a Chemical Master Equation [Van Kampen, 2007]. The stochastic simulations were performed using the modified version (https://github.com/jamesscottbrown/cuda-sim) of the software tool cuda-sim [Zhou et al., 2011], which implements the Gillespie stochastic simulation algorithm. The deterministic simulations were performed in MATLAB using a built-in ordinary differential equation solver ODE15S. The parameter fitting was performed using non-linear least squares routine FIT in MATLAB.
Acknowledgements
The authors would like to thank Prof J.Keasling for providing a plasmid, Prof De la Peña for advising on the hammerhead ribozyme. This work is supported by UK’s Engineering and Physical Sciences (EPSRC) Grant EP/M002454/1.