It is worth noting that the ratio of unfolded protein in steady state given in (3) generally will be far from the equilibrium value. So we've done a little math to quantify our intuition that some kind of chaperone mechanism is needed when folding is slow, and equally importantly, set the stage for more realistic models. In the limit that unfolding is much slower than removal ($\ku \ll \kr$), the ratio approaches $\ka / \kf \gg 1$ reflecting the fractional outflows from unfolded state. Our re-write of the ratio shows that aggregation is indeed expected to be significant in our simple analysis without the presence of chaperones: even though the first term in the square brackets may be small due to slow unfolding (i.e., protein stability), it must be positive and hence the whole ratio must exceed $\ka / \kf$, which is large. For proteins that are slow to fold spontaneously, we expect that the aggregation rate $\ka$ is much larger than the folding rate $\kf$ this is, after all, why chaperones are needed in the first place. To solidify our understanding of this almost-but-not-quite trivial model, we can rewrite (4) as $(\ka / \kf) \, $. The result depends only on rate constants and not on the absolute concentrations, which makes it straightforward to interpret. That is, the net flow from U to F must match the flow that is removed:
This ratio is determined using the continuity of flow from the unfolded to folded to the "removed" state (upper right in figure above). Our mathematical task is simplified by the observation that the ratio (1) does not require the absolute values of the concentrations, but only their ratio.
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