Where the "tot" superscripts are omitted because $\kutot / \kftot = \ku / \kf$ in the case of passive chaperones. In other words, among the eight rate constants in the cycle, only seven can be considered as adjustable parameters due to the cycle constraint All we need to do is set up the math to figure out what happens.īefore getting into detailed analysis of the model, we immediately see that it contains a cycle (U-F-FC-UC), and therefore the rates must satisfy a constraint, as holds for all cycles. Hence, the net result of the model will be modified, "effective" rate constants that will replace $\kf$ and $\ku$ in our analyses above. To give away the punchline first, note that our new model adds to the prior model only by adding an additional pathway between the unfolded and folded state. However, because we are considering a driven non-equilibrium condition, the chaperones' presence can alter the aggregation ratio. The chaperones will act simply as catalysts.
#Chaperone proteins keep unfolded of free
Thus, perhaps ironically, the driving in this case shifts the populations toward the dangerous unfolded state, though this would appear to be intrinsic to the directionality of the system - proteins start out unfolded! The simplest chaperone model - no ATPĪlthough this model is more complicated than our previous one, it has the distinct advantage of actually including chaperones! Note that the chaperones are purely "passive" in the model as shown - they store no free energy and do not use ATP. The balance condition which must hold in equilibrium would dictate a ratio of $\ku / \kf$, which differs significantly from (3) given our assumption that $\ku$ is small compared to other rates. 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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