Overview
In order to explore the approximate shape of ELP aggregates and the influence of temperature on ELP aggregation, we established the ELP-Temperature-Control Model
Figure 1: 3D Structure of ELP
Based on the Aggregation Model and Temperature Control Function, we modified the fractal dimension D, concentration, ELP chain length and other parameters. It is worth mentioning that we have performed an exponential correction to the temperature control function, and solved the limitation of the aggregation model for ELP, and then we got the final model $$ R(i)=R_p\cdot(\frac{A\cdot i}{S})^\frac{1}{D}\cdot \max(0,\ {-1+e}^{T_t-T_{tc}-K}) $$ What surprised us was that we not only proved the temperature control sensitivity of ELP and the feasibility of aggregation, but also provided strong support for the phenomenon that the aggregation is an ellipsoid from a mathematical point of view. Meanwhile, we found that it also has a perfect predictive effect when using this model to predict the temperature-controlled aggregation of ELP of other chain lengths
Process of Building ELP Aggregation Model
The results of the dynamic light scattering we did tell us that at \(\mathrm{44^\circ\mathrm C}\), the size of the ELP aggregation increases sharply, and the \(\mathrm{\zeta}\) potential at \(\mathrm{44^\circ\mathrm C}\) also showed an inflection point at \(\mathrm{44^\circ\mathrm C}\). To explain it, We established this model.
Model Assumptions
A single ELP is a spherical substance;
The number of ELP repetitions is small, that is, the length is short; in the solution, it is assumed to be a spherical substance.
ELP gathers into a spheroidal aggregation;
It becomes a sphere or sphere-like body after aggregation, that is, the surface area can be calculated as a sphere or ellipsoid
Aggregation is not affected by concentration.
For a single ELP aggregation model, we assume that it is not affected by concentration-that is, no growth after aggregation is completed
Two Possible Models
We've tried the aggregation model and the growth model, and our final model was established according to them.
1 Aggregation-Model
Aggregation model doesn't take temperature into consideration.
$$ R(i)=R_p\cdot(\frac{A\cdot\ i}S)^\frac1D $$
\(\mathrm{R(i)}\) is the radius of an aggregate;
\(\mathrm{R_p}\) is radius of a single protein;
\(\mathrm{i}\) is the number of protein molecules;
\(\mathrm{A}\) is a coefficient related to environment condition;
\(\mathrm{S}\) is the superficial area of aggregate;
\(\mathrm{D}\) is the fractal dimension. \(\mathrm{D=1}\) when a geometry is a line;\(\mathrm{D=\frac13}\) when a geometry is a sphere.
When \(\mathrm{A=1}\):
Figure 2: A=1
When \(\mathrm{D=\frac13}\):
Figure 3: D=1/3
2 Growth-Model
This model takes concentration into account.
$$ \frac{\mathrm dR}{\mathrm dt}=v(R)\cdot\frac{\big[\frac{c_0}{C-c_0}\big]R+R_c}{R-R_c} $$
\(\mathrm{R}\) is the radius of an aggregate;
\(\mathrm{c_0}\) is the initial concentration of ELPs;
\(\mathrm{R_c}\) is the maximum radius of aggregate;
\(\mathrm{C}\) is the concentration of ELP when radius arrives \(\mathrm{R_c}\);
\(\mathrm{v(R)}\) is a coefficient, which we assign to -1 for the assumption that environment is ideal.
We added coefficient \(\mathrm{\ln kt}\), in which \(\mathrm{k}\) is assigned to 0.001, reflecting that concentration varies with time.
$$ \frac{\mathrm dR}{\mathrm dt}=v(R)\cdot\frac{\big[\frac{c_0}{(C-c_0)\cdot\ln kt}\big]R+R_c}{R-R_c} $$
Figure 4: Radius-Time for different model
The Final Model
Combining the growth model and Aggregation-Model, we found that as long as the ELP number is sufficient and the time is long enough for a single aggregation model, the constraint radius can theoretically grow indefinitely. In other words, for any set constraint radius, as long as the local concentration is large enough, which means the number of ELP is sufficient , the stable theoretical radius achieved in a long enough time will be infinitely close to the constraint radius and fluctuate violently around this value.
Therefore, in order to make the results more intuitive, we preferred to analyze from the perspective of the number of ELP when the temperature factor is introduced.
So as to we chose the Aggregation-Model for subsequent corrections.
We first found this function to correct:
$$ \mathrm{T_t=T_{tc}+\frac{k}{\mathrm{length}}\cdot\ln{\frac{C_c}{\mathrm{conc}}}}
$$
To facilitate the trend of the model, we assumed that the correlation between temperature and concentration is minimal, so \(\mathrm{\ln{\frac{C_c}{\mathrm{conc}}}}\) was omitted. Therefore, we integrated the simplified temperature function with the original Aggregation-Model:
$$ \mathrm{R(i)=R_p(\frac{A\cdot i}{S})^{\frac1D}\cdot\max({0,T_t-T_{tc}-\frac k{length}}})
$$
- \(\mathrm{T_t}\) is the temperature variable
- \(\mathrm{T_{tc}}\) and \(\mathrm{k}\) are parameters inherent to each ELP
- \(\mathrm{length}\) is the chain length of the ELP
Figure 5: Diameter-Temperature
We noticed that this is not a good fit. After careful research, we found ELP is a temperature sensitive peptide, which means we should revise the temperature function to an exponential form:
$$ \mathrm{\exp({T_t-T_{tc}-\frac{k}{length}})=1}
$$
$$ \mathrm{R(i)=R_p\cdot(\frac{A\cdot i}{S})^\frac{1}{D}\cdot\ max{(0,{-1+e}^{T_t-T_{tc}-K})}}
$$
Figure 6: Diameter-Temperature
From this figure, we could see that although the trend of the model is consistent with that of the experimental data, the abrupt change of phase transition temperature is predicted by the model, which is still seriously inconsistent with the experimental data.
Looking at the original Aggregation-Model and the assumptions, we found: For this formula, although we took into account the sensitivity of ELP to temperature, we did not modify the original R.
In other words, the initial model was aimed at the aggregation of most non-temperature controlled proteins, so they would not show a particularly large trend with the increase of temperature. Therefore, for a specific number of peptides, if under certain conditions,they will have a dynamic theoretical aggregation diameter, so in the moment when the temperature factor is not zero, the model will show the aggregation effect of most proteins.
That is the aggregation diameter abrupt change, which is inconsistent with ELP.
To verify the above conjecture, we carefully observed the Aggregation-Model again.
Figure 7: Diameter-Number
It can be seen that when the number of ELP is \(\mathrm{10\times10^6}\), it has a diameter close to 1400nm when the temperature is not considered and the relative equilibrium is reached.
In order to verify that the abrupt change does not affect the trend, we increased the data of the experimental group by 1400nm, that is, we assumed that the initial aggregation diameter was 1400nm
Figure 8: Diameter-Temperature
It can be seen from this figure that the defects of the original Aggregation-Model and the correction conjecture are reasonable. That is, if the initial diameter of the ELP aggregate obtained in the experiment was 1400nm, the degree of fit is relatively high.
From above, we could see the rationality of the model, the reliability of the experimental data and the rationality of the corresponding correction ideas.
At last, we considered the effect of concentration. So as to we introduced \(\mathrm{K}\), which derived from the original temperature function and equals to \(\mathrm{k\cdot \ln \frac{C_c}{conc}}\), to describe the effect of concentration.
$$ \mathrm{R(i)=R_p\cdot(\frac{A\cdot i}{S})^\frac{1}{D}\cdot\ max{(0,{-1+e}^{T_t-T_{tc}-K})}}
$$
Figure 9: Diameter-Temperature
It can be found that the fitting degree is the best when K=3, but K=2 is more appropriate in terms of temperature sensitivity, but in fact, the value of K should be related to concentration and chain length.
It can be considered that for peptides sensitive to temperature control, such as ELP, when different chain lengths are selected but the control concentration factors are consistent, the K value will change exceedingly. The K value changed little for the protein aggregation with less temperature sensitivity.
So we got the final ELP Model:
$$ \mathrm{R(i)=R_p\cdot(\frac{A\cdot i}{S})^\frac{1}{D}\cdot\ max{(0,{-1+e}^{T_t-T_{tc}-K})}}
$$
\(\mathrm{i}\) represents the actual number of ELP;
\(\mathrm{A}\) represents the effect of the solution on the effective number: greater than 1 means promoting, less than 1 means weakening
\(\mathrm{S}\) is the surface area of the aggregate
\(\mathrm{R_p}\) represents the radius of ELP
\(\mathrm{T_t}\) is the actual temperature
\(\mathrm{T_{tc}}\) represents the phase transition temperature
\(\mathrm{K}\) represents the coefficient related to concentration and ELP chain length
\(\mathrm{D}\) stands for fractal dimension
D value Verification and Rational Interpretation of Experimental Data
1 Experimental group found that aggregates are not all spheres but mostly ellipses observed under AFM.
So, with this model, we wanted to explore the certain shape of the ELP aggregation:
Figure 10: Diameter-Temperature
Careful observation will reveal slight inconsistence with the previous hypothesis.
D, which is more consistent with the trend, should be between 1/2 and 2/3.That is to say, the shape of aggregation should be rather than a complete sphere, nor an ellipsoid with large eccentricity, but a spheroid
Figure 11: Diameter-Temperature
2 According to the above D, the value of K is modified again, and the result is: K=3.5 has the best degree of fit. And we could notice that the accumulated temperature abrupt-change point will be a little later than the actual data, about 3 degrees Celsius higher:
Figure 12: Diameter-Temperature
From above, we could reasonably explain:
A true aggregate is an ellipsoid with fractal dimension D between 1/2 and 2/3
The actual critical temperature is not the actual temperature began to gather, and after will be slightly a bit, that is to say, the experimental data of wobbles around 45 degrees Celsius is reasonable: ELP began to gather, but didn't really produce good aggregate, or compare adhesion, but the instrument when measuring the think they have already started to gather for tangible aggregates
For the second point, we could also verify it from the following figure
Figure 13: Diameter-Temperature
It can be found that for different fractal dimensions D, there will almost be an intersection point G (the corresponding temperature is \(\mathrm{T_g}\)) about 3 from the phase transition temperature, and in the experimental data, the temperature at which significant changes begin to occur (namely, the temperature at which significant changes occur in the curve gradient) is also about 3 from the actual aggregation temperature
In other words, the actual measured phase transition temperatures do begin to cluster, but do not tend to a particular shape.Due to the lack of energy acquisition, ELP aggregates are likely to be in an unstable aggregation-dispersion state, so they have different shapes, and any D cannot well represent the shape of a certain aggregate.
However , after the \(\mathrm{T_g}\), ELP gradually changes from the unstable aggregation-dispersion state to the aggregation-tendency state, and the measurement of its aggregation diameter is more accurate, so it will increase rapidly, and there is always a D that can be well fitted and predicted
Final Conclusion:
After several corrections, we found that K and D values need to be modified for ELP of different lengths, but the establishment of the model and the experimental data are reasonable and correct
In other words, the model can be used to predict the aggregation of ELP, and the uncertainty of experimental data is high, but it is still correct, or it fluctuates within a reasonable range.
Prediction of ELP with chain length=54
$$ \mathrm{R(i)=R_p\cdot(\frac{A\cdot i}{S})^\frac{1}{D}\cdot\ max{(0,{-1+e}^{T_t-T_{tc}-K})}}
$$
Obviously, all values except Ttc, K, Rp and D can be considered unchanged
Corresponding changes can be predicted from the analysis of temperature control sensitivity of ELP:
- \(\mathrm{R_p}\) will be larger.
- \(\mathrm{D}\) will be between 1/2 and 1, probably between 1/2 and 2/3
- \(\mathrm{K}\) value will increase, because the longer the ELP, the more sensitive it is to temperature
- \(\mathrm{T_{tc}}\) may be smaller, because for ELP, the longer it is, the higher the aggregating tendency will be. At lower temperature, we can consider it as the beginning of aggregation or the tendency of aggregation
Therefore, we could obtain the following model prediction graph
Figure 14: Diameter-Temperature
Using interpolation, we could see a better prediction:
Figure 15: Diameter-Temperature
The prediction of experimental data is based on the understanding of experimental data with Length =27 and ELP, that is, the phase transition temperature decreases
Therefore, the prediction effect is perfect.
In addition, it was found that if the protein without considering the temperature, its aggregation diameter would reach 1400nm after the length became longer, rather than close to 1400nm like length=27 without reaching 1400nm, indicating the rationality of the model
Applicability of the model
Although the model can explain the existing data well, whether it is universal is questionable
For this model, we believe that:
- It is difficult to predict proteins that are not sensitive to temperature control, or the predicted results should differ greatly from the actual values
- For some proteins whose aggregates are strongly affected by concentration, this model is not applicable;Because the model is built on the basis of a single aggregate and relatively little affected by concentration
- We could not well explain the proteins whose growth factor was greater than aggregation factor, because our model was based on aggregation model, and we abandoned the growth model
But the model is still applicable:
- For ELP itself, we will find that it has a good predictive and explanatory effect
- The shape of the aggregation can be well predicted.D value will obviously change the clustering tendency, that is to say, D value has a great influence on the model, so the clustering shape can be predicted by experimental data and model
- It can be well explained and fitted for proteins with different aggregation shapes
- for the aggregation factor is greater than the growth factor, the influence of number is greater than the influence of concentration of the experiment, can be well fitted and explained
Conclusion
Congratulation!
The ELP Temperature-Controlled Aggregation Model we have established is very effective, and it has completed all the content beyond expectations.
- The experimental phenomenon that the aggregate is an ellipsoid is verified, and the theoretical support is given that the aggregate is an ellipsoid
- The theoretical basis for the strong correlation between the ELP chain length change and the temperature change was verified, and the theoretical support for the strong correlation between the ELP chain length change and the temperature change was given.
- The prediction and theoretical support of the actual aggregation forming temperature of the ELP are given
- The prediction of the ELP aggregation of different chain lengths is given
References
[1]He P. Kinetics Modelling and Doctor Degree in Engineering[D]. Harbin University of Science and Technology, 2013.
[2]Zhou Y, Wang TT, Yan DD, et al. Advances in Biotechnological Application of Elastin-like Polypeptides as Functional Nanomaterials[J]. Biotechnology Bulletin, 2020,36(11):198-208.
[3]Zhang GY, Chen ZS, Li CH, et al. Molecular dynamics of elastin-like peptides[J]. Computers and Applied Chemistry, 2011,28(04):399-402.