Team:Harvard/Model
Pathway Model
1 Overview
Curli is the main proteinaceous component of the extracellular matrix naturally produced by E. coli. Although the main structural component is the self-assembling csgA monomer, there are a number of other proteins involved in its production and export. The curli production pathway can be broken down into two main modules: gene expression and translocation. The first, gene expression, is comprised of transcription and translation. Translocation can be further broken down into periplasmic export and extracellular secretion.
Spatial parameters, heat and diffusion gradients, etc. will not be considered in this model to avoid partial differential equations.
2 Gene Expression
2.1 Transcription
The naturally occurring genes corresponding to curli are organized into csgBAC and csgDEFG operons. csgD is a regulatory protein, and thus is not applicable for the expression of foreign plasmids. Additionally, due to the additional protein domains fused to the csgA monomer, it is placed on a plasmid on its own (\(g_{csgA}\)). The remaining genes are placed on another plasmid (\(g_{csgCEFG}\)). The rate of transcription is primarily governed by the plasmid copy number and promoter strength. Although RNA polymerase and a number of other transcription factors are also involved in the transcription of mRNA, transcription factor binding achieves equilibrium much faster than transcription, translation, and protein accumulation, so it can be at considered to be at steady state on the time scale of proteins. Thus only the concentration of the plasmids are assumed to have a large impact on the rate of transcription. $$g_{csgA} \overset{\alpha_{1}}{\rightarrow} g_{csgA} + mRNA_{csgA}$$ $$g_{csgBCEFG} \overset{\alpha_{2}}{\rightarrow} g_{csgBCEFG} + mRNA_{csgBCEFG}$$ The rate of transcript degradation is dependent upon transcript stability, which is in turn affected by a number of factors. Although we have introduced these genes on foreign plasmids, the mRNA half-life here is assumed to be that of the mRNA transcripts of the corresponding genes naturally present in E. coli. $$mRNA_{csgA} \overset{\zeta_{1}}{\rightarrow} \varnothing$$ $$mRNA_{csgBCEFG} \overset{\zeta_{1}}{\rightarrow} \varnothing$$ The degradation rate, \(\zeta_{n}\), can be described as an exponential decay function of time mRNA half-life, \(h_n\): $$\zeta_{n} = e^{h_nt}$$
2.2 Translation
Each of the coding sequences in the two transcripts described above are preceded by an RBS sequence that determines the rate of translation of each protein. The relative RBS strengths determine the stoichiometry between the proteins involved in the curli pathway. Translation also involves a number of players, including the ribosome and tRNAs, but as the kinetic parameters regarding the rate of translation available in literature are based $$mRNA_{csgA} \overset{\beta_{1}}{\rightarrow} mRNA_{csgA} + csgA_{cyt}$$ $$mRNA_{csgBCEFG} \overset{\beta_{2}}{\rightarrow} mRNA_{csgBCEFG} + csgB_{cyt}$$ $$mRNA_{csgBCEFG} \overset{\beta_{3}}{\rightarrow} mRNA_{csgBCEFG} + csgC_{cyt}$$ $$mRNA_{csgBCEFG} \overset{\beta_{4}}{\rightarrow} mRNA_{csgBCEFG} + csgE_{cyt}$$ $$mRNA_{csgBCEFG} \overset{\beta_{5}}{\rightarrow} mRNA_{csgBCEFG} + csgF_{cyt}$$ $$mRNA_{csgBCEFG} \overset{\beta_{6}}{\rightarrow} mRNA_{csgBCEFG} + csgG_{cyt}$$ $$csgA_{cyt} \overset{\zeta_{3}}{\rightarrow} \varnothing$$ $$csgB_{cyt} \overset{\zeta_{4}}{\rightarrow} \varnothing$$ $$csgC_{cyt} \overset{\zeta_{5}}{\rightarrow} \varnothing$$ $$csgE_{cyt} \overset{\zeta_{6}}{\rightarrow} \varnothing$$ $$csgF_{cyt} \overset{\zeta_{7}}{\rightarrow} \varnothing$$ $$csgG_{cyt} \overset{\zeta_{8}}{\rightarrow} \varnothing$$ The degradation rates of the proteins can be described by the same exponential decay function for mRNA transcripts above.
3 Translocation
Although the ultimate destination of the csgA monomer is the extracellular space, not all of the other proteins involved in the pathway have the same fate. csgB and csgF do operate in the extracellular space, but csgC and csgE are chaperone proteins that remain in the periplasm, whereas csgG forms a channel in the outer membrane.
3.1 Periplasmic export
Since none of the curli proteins remain in the cytoplasm, all must translocate into the cell's periplasm. The mechanism by which this occurs is the Sec secretion pathway. The main actors in this secretion pathway are SecYEG, the protein conducting channel (PCC), SecA which acts as an ATPase driving the translocation, and SecB, a chaperone protein that keeps proteins in an unfolded state (Driessen et al., 2007). As a protein emerges from the ribosome, SecB, a homotetramer, binds and stabilizes it in its unfolded conformation. SecB binds to SecA, a homodimer which also recruits SecYEG to assemble a dimeric PCC. $$csgA_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgA_{per}$$ $$csgB_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgB_{per}$$ $$csgC_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgC_{per}$$ $$csgE_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgE_{per}$$ $$csgF_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgF_{per}$$ $$csgG_{cyt} + 4 SecB + 2 SecA + 2 SecYEG \overset{\gamma_{1}}{\rightarrow} 4 SecB + 2 SecA + SecYEG + csgG_{per}$$ The rate of binding and secretion is assumed to be conserved across the different proteins due to the similarity in mechanism and homology in the Sec signal sequences. $$csgA_{per} \overset{\zeta_{9}}{\rightarrow} \varnothing$$ $$csgB_{per} \overset{\zeta_{10}}{\rightarrow} \varnothing$$ $$csgC_{per} \overset{\zeta_{11}}{\rightarrow} \varnothing$$ $$csgE_{per} \overset{\zeta_{12}}{\rightarrow} \varnothing$$ $$csgF_{per} \overset{\zeta_{13}}{\rightarrow} \varnothing$$ $$csgG_{per} \overset{\zeta_{14}}{\rightarrow} \varnothing$$
3.2 Extracellular secretion
Analysis of the crystal structure of csgG and has revealed that it assembles into a double-nonameric form in D9 symmetry (Taylor and Matthews 2015). CsgE has also been shown to form a nonamer at the base of the csgG structure in the periplasm, providing selectivity for the substrates that are secreted (Goyal et al., 2014). CsgG and csgE participate in the translocation fo csgF into the extracellular matrix, which then folds and binds csgG to the membrane. Meanwhile, csgC interacts with csgA and csgB monomers to prevent the formation of oligomers (Taylor and Matthews 2015). When the monomers interact with csgE, they become trapped in the periplasmic cavity and are transported across the outer membrane. CsgB then interacts with csgF to initiate the nucleation of csgA fibers. $$9 csgG_{per} \underset{\delta_{-1}}{\overset{\delta_{1}}{\rightleftharpoons}} csgG_{9}$$ $$9 csgE_{per} \underset{\delta_{-2}}{\overset{\delta_{2}}{\rightleftharpoons}} csgE_{9}$$ $$2 csgF_{per} \overset{\delta_{3}}{\rightarrow} 2 csgF_{ECM}$$ $$ csgG_{9} + 2 csgF_{ECM} + csgE_{9} \overset{\delta_{4}}{\rightarrow} csgGEF$$ $$csgA_{per} + csgGEF \overset{\delta_{5}}{\rightarrow} csgGEF + csgA_{ECM}$$ $$csgB_{per} + csgGEF \overset{\delta_{6}}{\rightarrow} csgGEF + csgB_{ECM}$$ However, as kinetic data regarding the assembly of the multimers are not available, we will simplify the reactions above into the following: $$2 csgF_{per} \overset{\delta_{3}}{\rightarrow} 2 csgF_{ECM}$$ $$csgA_{per} + 9 csgG_{per} + 9 csgE_{per} + 2csgF_{ECM} + csgC \overset{\delta_{5}}{\rightarrow} 9 csgG_{per} + 9 csE_{per} + 2csgF_{ECM} + csgC + csgA_{ECM}$$ $$csgB_{per} + csgGEF + csgC \overset{\delta_{6}}{\rightarrow} csgGEF + csgC + csgB_{ECM}$$ As we are not concerned with the degradation of the proteins once they have been secreted from the cell, we will not include those reactions here.
4 Aggregation and Polymerization
5 Differential Equations
Transcription
$$\frac{d[mRNA_{csgA}]}{dt} = \alpha_{1}[g_{csgA}] - \zeta_{1}[mRNA_{csgA}]$$ $$\frac{d[mRNA_{csgBCEFG}]}{dt} = \alpha_{2}[g_{csgBCEFG}] - \zeta_{2}[mRNA_{csgA}]$$
Translation
$$\frac{d[csgA_{cyt}]}{dt} = \beta_{1}[mRNA_{csgA}] - \zeta_{3}[csgA_{cyt}]$$ $$\frac{d[csgB_{cyt}]}{dt} = \beta_{2}[mRNA_{csgA}] - \zeta_{4}[csgA_{cyt}]$$ $$\frac{d[csgC_{cyt}]}{dt} = \beta_{3}[mRNA_{csgA}] - \zeta_{5}[csgA_{cyt}]$$ $$\frac{d[csgE_{cyt}]}{dt} = \beta_{4}[mRNA_{csgA}] - \zeta_{6}[csgA_{cyt}]$$ $$\frac{d[csgF_{cyt}]}{dt} = \beta_{5}[mRNA_{csgA}] - \zeta_{7}[csgA_{cyt}]$$ $$\frac{d[csgG_{cyt}]}{dt} = \beta_{6}[mRNA_{csgA}] - \zeta_{8}[csgA_{cyt}]$$
Periplasmic Export
$$\frac{d[csgA_{per}]}{dt} = \gamma_{1}[csgA_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{9}[csgA_{per}]$$ $$\frac{d[csgB_{per}]}{dt} = \gamma_{1}[csgB_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{10}[csgB_{per}]$$ $$\frac{d[csgC_{per}]}{dt} = \gamma_{1}[csgC_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{11}[csgC_{per}]$$ $$\frac{d[csgE_{per}]}{dt} = \gamma_{1}[csgE_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{12}[csgE_{per}]$$ $$\frac{d[csgF_{per}]}{dt} = \gamma_{1}[csgF_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{13}[csgF_{per}]$$ $$\frac{d[csgG_{per}]}{dt} = \gamma_{1}[csgG_{cyt}][SecB]^4[SecA]^2[SecYEG]^2 - \zeta_{14}[csgG_{per}]$$
Extracellular Secretion
$$\frac{d[csgF_{ECM}]}{dt} = \delta_{3}[csgF_{per}]$$ $$\frac{d[csgA_{ECM}]}{dt} = \delta_{5}[csgA_{per}][csgG_{per}]^9[csgE_{per}]^9[csgF_{ECM}]^2[csgC]$$ $$\frac{d[csgB_{ECM}]}{dt} = \delta_{5}[csgA_{per}][csgG_{per}]^9[csgE_{per}]^9[csgF_{ECM}]^2[csgC]$$
Aggregation and Polymerization
6 Rate constants
| Symbol | Definition | Value | Units | Reference | \(\alpha_1\) | Rate of transcription of csgA | TBD | TBD | TBD |
|---|
