# Diffusion-Enhanced Förster Resonance Energy Transfer in Flexible Peptides: From the Haas-Steinberg Partial Differential Equation to a Closed Analytical Expression

^{*}

## Abstract

**:**

## 1. Introduction

_{T}(r), with the rate constant of deactivation in the donor-only peptide, k

_{D}, with the Förster distance, R

_{0}, a constant that characterizes the donor-acceptor pair, and with the donor–acceptor distance, R

_{DA}.

_{rad}(“k

_{radiative}”); by being quenched, for instance by iodine ions, with the rate constant k

_{nrad}(“k

_{non-radiative}”); or by transferring its excitation energy to the acceptor (A) with the rate constant k

_{FRET}. In experimental measurements, we always use the donor-acceptor peptide (the DA-peptide) together with the “donor-only” peptide (the D-peptide). If no acceptor is present but only the donor, the measured rate of deactivation is k

_{D}= k

_{rad}+ k

_{nrad}(Figure 1; (I)). In the donor-acceptor peptide, in presence of an acceptor, the measured rate is k

_{DA}= k

_{rad}+ k

_{nrad}+ k

_{FRET}(Figure 1, (II)). Subtracting (I) from (II) yields the effective FRET rate, k

_{FRET}, k

_{FRET}= k

_{DA}− k

_{D}(Equation (2)). In any normal situation, k

_{FRET}increases with increasing diffusional motion between donor and acceptor. During the lifetime of the donor fluorescence, donor and acceptor can approach each other, and the probability of the FRET event increases in accordance with Förster’s law (Equation (1)), where k

_{T}is the distance-dependent Förster transfer rate constant at a specific donor–acceptor distance, R

_{DA}. Consequently, k

_{FRET}increases with diffusion, and the effective distance derived from k

_{FRET}(Figure 1, (III), (IV), Equations (3) and (4)) decreases. In summary, the effective distance (Equation (4)) is a suitable diagnostic parameter to measure the diffusion enhancement of FRET. The effective distance serves as the interface between experiment and theory, that is, between the experimental results and the numerical solutions of the Haas-Steinberg equation (HSE). To restate, the effective distance is obtained experimentally from steady-state or time-resolved measurements on the peptide equipped with donor and acceptor, the “DA-peptide” (see Figure 1), and the peptide equipped only with the donor, the “D-peptide”. Measurements on the DA-peptide yield the fluorescence intensity, I

_{DA}, in steady-state experiments or the fluorescence decay rate, k

_{DA}, in time-resolved experiments, while measurements on the D-peptide yield I

_{D}or k

_{D}. Exemplary chemical structures (Figure S1) and kinetics (Figure S2a,b) are given in the Supporting Information, Chapter 3.

#### Introduction to the Haas-Steinberg Equation

## 2. Materials and Methods

#### 2.1. The HSE

_{D}, instead of the donor deactivation rate constant, k

_{D}, used in Figure 1 (Equation (I): k

_{D}= k

_{rad}+ k

_{nrad}). It is much easier to recall the value of a lifetime, for instance, 100 ns, than that of a rate constant, for instance, 0.01 ns

^{−1}. The donor lifetime in the donor-only peptide, τ

_{D}, is simply the inverse of the rate constant, k

_{D}, in this peptide (Equation (6)).

_{0}= N(r, t = 0) in term-(3) is the initial probability density distance distribution. Upon irradiation, donors become activated with a probability assumed to be random, to be independent of the presence of an acceptor at various distances. In consequence, N

_{0}is identical to the equilibrium distance distribution, p(r), of all chains present in the measurement sample. This distance distribution, p(r), as well as the site-to-site diffusion coefficient, D, in term-(3), is what we are interested in. They inform us on the structure and the dynamics of the (bio)polymer under investigation.

_{D}, and R

_{0}. The HSE relates the rate of donor deactivation, term-(0); to the rate of donor deactivation in the absence of FRET, term-(1); plus the additional rate due to FRET, term-(2); plus the additional rate due to diffusion enhanced FRET, term-(3). A numerically obtained solution of the HSE yields the value of the effective distance between donor and acceptor. We elaborate on this calculation in Supporting Information, Chapter 4, Figure S3a–i.

#### 2.2. Steinberg’s Derivation

_{eff}from experiments as well as from numerically solving the HSE. We can therefore use the HSE to globally fit the experimental data. This was quite arduous in the past, where the HSE served a bit as a black box [5,6,10,11,12,13,14]. In the Results Section, we develop the Haas-Steinberg–Jacob equation (HSJE), a closed analytical expression (CAE). Any global fit based not on the HSE (PDE) but on the HSJE (CAE) is faster by about two orders of magnitude.

## 3. Results

#### 3.1. Ideas and Concepts

_{D}and R

_{0}, we solved the HSE for a large range of values of the diffusion coefficient reaching from 10

^{−3}to 10

^{6}Å

^{2}/ns. The obtained effective-distance values formed a full diffusion profile, ranging from the “static limit” in the near absence of diffusion to the “dynamic limit” at high diffusion. In figures and graphs, we always plotted the effective distance against the square root of the diffusion coefficient or, later, against the square root of the “augmented diffusion coefficient” to keep the x-axis numbers or labels manageably small and readable (Figure 2b).

_{L}, to the right-integration limit, r

_{R}. A physical but oversimplified perspective is to view r

_{L}as the closest possible distance between donor and acceptor.

_{eff}approaches L, to 1 or 100%, when R

_{eff}approaches R (Figure 2c). Thus, we distinguish in Figure 2 between the diffusion profiles, the R

_{eff}(D) or R

_{eff}(D

^{1/2}) profiles (Figure 2b), and the diffusion-influence profiles, the DI(D) or DI(D

^{1/2}) profiles (Figure 2c), or the DI(J), DI(J

^{1/2}), or DI(X) profiles (Figure 2d) after we discovered that the decisive independent variable, the decisive x-variable, is not D but J or its square root, X (Equation (12)).

^{1/2}, as diffusion-influence profiles, as DI, DI(J), or DI(X) profiles.

^{0}= J

^{1/2}. The value X

^{0}is the X value where DI = 50%; the M value describes the steepness of the sigmoid.

_{0}, 50%) provided that a logarithmic scale is chosen for the x-axes or, equivalently, that DI is plotted against lnX. The numerically obtained profiles could then be perfectly fitted according to Equation (13). Thus, we are convinced that any DI profile follows a symmetrical sigmoid in accordance with Equation (13). This requirement of sigmoidal symmetry served as the second important control that indicates correctly performed numerical simulations. Equations (12) and (13), in combination, yield Equation (14), a first raw form of the HSJE.

_{0}and M in Equation (13) are functions of just a single variable, of the ratio of the Förster radius and the left-integration limit, of R

_{0}/r

_{L}. Thus, in the last step, we completed the HSJE by numerical work to determine how X

_{0}and M depend on R

_{0}/r

_{L}.

#### 3.2. The Equivalence Statements

^{(1)}and HSE

^{(2)}, where HSE

^{(1)}uses set

^{(1)}that consists of the distance distribution, p

^{(1)}(r), and the values, D

^{(1)}, τ

_{D}

^{(1)}, R

_{0}

^{(1)}, and r

_{L}

^{(1)}, while HSE

^{(2)}uses set

^{(2)}that consists of the distance distribution, p

^{(2)}(r), and the values, D

^{(2)}, τ

_{D}

^{(2)}, R

_{0}

^{(2)}, and r

_{L}

^{(2)}. Accordingly, the augmented diffusion coefficient J

^{(1)}equals D

^{(1)}τ

_{D}

^{(1)}/(R

_{0}

^{(1)})

^{2}and J

^{(2)}equals D

^{(2)}τ

_{D}

^{(2)}/(R

_{0}

^{(2)})

^{2}. We also have to distinguish between the two diffusion-influence profiles obtained from HSE

^{(1)}and HSE

^{(2)}, between DI

^{(1)}and DI

^{(2)}as functions of J or X, as they usually stem from different distance distributions, p

^{(1)}(r) and p

^{(2)}(r), that usually come with different L- and R-values, L

^{(1)}and R

^{(1)}and L

^{(2)}and R

^{(2)}(see Equations (9)–(11)).

#### 3.2.1. The First Equivalence Statement: Variation of the Distance Distribution

#### Equivalence Statement 1 (ES1, Text)

^{(1)}(r) and p

^{(2)}(r), coincide if they were obtained with the same values of the donor lifetime, τ

_{D}, the Förster radius, R

_{0}, and the left-integration limit, r

_{L}.”

^{(1)}and J

^{(2)}are obviously identical, and using J as the independent variable makes no difference versus using D instead, but it fosters the formal parallelism of the four equivalence statements. Two questions arise:

- 1.
- Why do we demand that r
_{L}^{(1)}= r_{L}^{(2)}? Why can ES1 not be applied if the left-integration limit differs in both sets?

_{L}values, FRET becomes faster. As a consequence, higher values of the augmented diffusion coefficient are necessary to reach the dynamic limit in the diffusion-influence profiles. Therefore, the distribution with smaller r

_{L}will lead to a diffusion-influence profile that only later reaches DI values close to 100%. This is further detailed and illustrated in the Supporting Information, Chapter 7b, Figure S4a–c.

- 2.
- Is ES1 valid for any distance distribution equation or model? Or is it only valid for “well-behaved” models?

_{L}, that is, in the case of large jumps of probability-density values from zero to finite values at this point (Supporting Information, Chapter 7c, Figure S5a). In the transition region of the sigmoidal DI-profiles of two such distributions, the DI-value disagreement can become as large as 10%, undermining our efforts to develop a closed analytic equation with precisely determined constants, the HSJE. Further, such discontinuities can hardly be justified by any physical reasoning. This problem was resolved, and complete overlap was restored for all cases when we modified the distribution equations to display continuity of probability-density values at and around r = r

_{L}(Equations (18) and (19)), (Supporting Information, Chapter 7c, Figure S5b). Thus, we proceeded by using only “well-behaved” distribution equations that, by their very nature, guarantee that p(r = r

_{L}) = 0 and guarantee p(r)-value continuity (Equations (18) and (19)).

#### 3.2.2. The Second Equivalence Statement: Variation of the Donor Lifetime

_{D}as the independent variable in diffusion-influence plots is the first of two steps towards the augmented diffusion coefficient, J.

^{(1)}and set

^{(2)}, that only differ by the diffusion coefficient, D

^{(1)}in contrast to D

^{(2)}, and by the donor lifetime, τ

_{D}

^{(1)}in contrast to τ

_{D}

^{(2)}, are identical as long as the product D⋅τ

_{D}does not change, that is, as long as it is valid that D

^{(1)}τ

_{D}

^{(1)}= D

^{(2)}τ

_{D}

^{(2)}. We now formulate the second equivalence statement and prove it to be a mathematical consequence of the HSE.

#### Equivalence Statement 2 (ES2, Text)

^{(1)}to D

^{(2)}and the donor lifetime from τ

_{D}

^{(1)}to τ

_{D}

^{(2)}, but the product of diffusion coefficient and donor lifetime remains constant, then the diffusion-influence values obtained from the diffusion-influence functions, DI

^{(1)}(J

^{(1)}) and DI

^{(2)}(J

^{(2)}), are identical. The corresponding diffusion-influence profiles coincide”.

_{D}, appears in the rescaled HSE (Equation (23)) at the same place as D does in the initial HSE (Equation (5)). Secondly, neither D nor τ

_{D}appear at any other place of the rescaled HSE (Equation (23)). It is obvious from Equation (23) that any change of the diffusion coefficient and donor lifetime will not change the resulting effective distance, R

_{eff}, if their product, D⋅τ

_{D}, does not change. This is what ES2 expresses.

#### 3.2.3. The Third Equivalence Statement: Variation of the Förster Radius and Left-Integration Limit

_{0}

^{(1)}to R

_{0}

^{(2)}and the left-integration value from r

_{L}

^{(1)}to r

_{L}

^{(2)}. We observe total overlap of the resulting diffusion-influence profiles, DI

^{(1)}(J

^{(1)}) and DI

^{(2)}(J

^{(2)}), if the third equivalence statement is obeyed:

#### Equivalence Statement 3 (ES3, Text)

^{(1)}(J

^{(1)}) and DI

^{(2)}(J

^{(2)}), are identical. The corresponding diffusion-influence profiles coincide”.

_{0}-and r

_{L}-values where R

_{0}/r

_{L}was always held constant at 3. The three corresponding DI-profiles coincide (Figure 5c, X = J

^{1/2}).

_{R}, as simulations showed that r

_{R}has only to be chosen to be sufficiently large to enclose the distance distribution up to probability-density values close to zero. Additionally, the changing shape of the distribution caused by rescaling, its expansion or contraction in the x-direction, is of no consequence, as has been demonstrated (ES1, Figure 3 and Figure 6). Nevertheless, we keep in mind, in the ongoing analysis, the validity of ES3 rests on the validity of ES1. After rescaling, in accordance with Equation (26), the new left-integration limit follows Equation (29).

_{0}

^{(1)}to R

_{0}

^{(2)}and r

_{L}

^{(1)}to r

_{L}

^{(2)}and apply space rescaling to the corresponding HSE equations, HSE

^{(1)}and HSE

^{(2)}, we obtain J

^{(1)}and J

^{(2)}as well as s

_{L}

^{(1)}and s

_{L}

^{(2)}. Only if s

_{L}

^{(1)}and s

_{L}

^{(2)}are identical can we expect that the diffusion-influence values DI

^{(1)}(J

^{(1)}) and DI

^{(2)}(J

^{(2)}) and corresponding profiles coincide. According to Equation (29), this will be the case if r

_{L}

^{(1)}/R

_{0}

^{(1)}= r

_{L}

^{(2)}/R

_{0}

^{(2)}. This is equivalent to the condition used in ES3: R

_{0}

^{(1)}/r

_{L}

^{(1)}= R

_{0}

^{(2)}/r

_{L}

^{(2)}.

#### 3.2.4. The Fourth Equivalence Statement

#### Equivalence Statement 4 (ES4, Text)

^{(1)}(r) and p

^{(2)}(r): If the diffusion coefficient or the donor lifetime or the Förster radius are varied but not the augmented diffusion coefficient J composed of these parameters, and if the ratio of the Förster radius and left-integration limit is not varied, then the diffusion-influence values obtained from the diffusion-influence functions, DI

^{(1)}(J

^{(1)}) and DI

^{(2)}(J

^{(2)}), are identical. The corresponding diffusion-influence profiles coincide.”

_{0}/r

_{L}ratio corresponds to a single sigmoidal curve, a single diffusion-influence profile. We slide along that sigmoid when we vary J, that is (see Equation (11)), when we vary the diffusion coefficient, or the donor lifetime, or the Förster radius and the left-integration limit, but latter two only in combinations that keep R

_{0}/r

_{L}constant. Such a sigmoid follows Equation (13) with constant coefficients, X

_{0}and M.

_{0}/r

_{L}ratio is fixed, X

_{0}and M are constants that can be determined by fitting the numerically obtained sigmoid to Equation (13). As soon as we vary R

_{0}/r

_{L}, we obtain a different sigmoid characterized by different X

_{0}- and M-values. Thus, X

_{0}- and M are functions of R

_{0}/r

_{L}. We need to determine these functions to establish Equation (13) as an equation that covers all possible diffusion-influence profiles. Being content to firstly cover only a small two-dimensional range of R

_{0}and r

_{L}, we learned that these functions are then simple polynomials of second degree. In the following, the focus shifts from mainly mathematical reasoning to the need to obtain reliable results from numerical HSE-solutions.

#### 3.3. The Grid

_{0}and of 1.5–5 Å for r

_{L}: These are the crucial R

_{0}- and r

_{L}-ranges for short-distance FRET methods applied to flexible peptides or polymers [13,17,18] (see Supporting Information, Chapter 3, Figures S1 and S2). The R

_{0},r

_{L}-grid, or R

_{0},r

_{L}-set, was R

_{0}/Å × r

_{L}/Å = {9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0} × {1.5, 2.0, 2.5, 3.0, 3.5, 4,0, 4.5, 5.0} resulting in a total of 56 R

_{0},r

_{L}-combination. For each combination, we determined the corresponding DI-profile by numerically solving the HSE for at least 28 values of the diffusion coefficient. We then determined the X

_{0}- and M-value of each of these 56 profiles by fitting them to Equation (13). Indeed, the X

_{0}- and M-values coincided for all those R

_{0},r

_{L}-combinations with a constant ratio R

_{0}/r

_{L}. A selection of results, those for R

_{0}/r

_{L}equal to 3, 4, 5, or 6, are shown in Table 1. These results clearly corroborate ES3 and, by that, also ES1, on which ES3 rests.

_{0}- and M-values from fits of the numerically obtained sigmoids required perfectly symmetrical sigmoids in the first place. Recall, that we argued in the first chapter of Results, “Ideas and Concepts”, that any diffusion-influence profile should be a perfectly symmetrical sigmoid, a condition that when met points to a properly executed numerical simulation. A consequence of the normalization Equation (11) is that sigmoids numerically obtained from the HSE follow Equation (13) less accurately when the difference between the extreme values of the effective distance, L and R, when the amplitude of the sigmoid, ∆R

_{eff}= L − R, becomes smaller. Very narrow distance distributions only allow for accordingly small ∆R

_{eff}-values. It is then almost impossible to numerically obtain well-shaped sigmoids. Luckily, we were free to choose the distance distributions in any way we wanted (ES1) to evaluate the grid. We have chosen the simplest model possible, the ideal-chain or random coil model (Equation (18)). Here, the broadness of the distribution is determined by the parameter, b. Thus, to always guarantee broad distributions and to always guarantee a large difference, ∆R

_{eff}= L − R, when R

_{0}is varied within the grid, the parameter b was chosen to follow the equation b = 2⋅R

_{0}. An exemplary distribution (R

_{0}= 10 Å, r

_{L}= 3 Å) of this series is given by the black curve in Figure 6a.

_{0}and b = 1.5⋅R

_{0}. An exemplary distribution (R

_{0}= 10 Å, r

_{L}= 3 Å) of the latter is given by the blue curve in Figure 6a. Figure 6b compares the diffusion-influence profiles for R

_{0}= 10 Å, r

_{L}= 3 Å, b = 2⋅R

_{0}(black curve) and for R

_{0}= 10 Å, r

_{L}= 3 Å, b = 1.5⋅R

_{0}(blue curve). Both profiles overlap so well that visual distinction is impossible. This is valid for comparisons at any of the employed R

_{0}-values as is shown in Figure 6c. The M-values were virtually identical, and the X

_{0}-values (lower black and blue data points) were always so close that the corresponding DI-profiles were visually indistinguishable (see Figure 6b).

_{0}are sufficiently broad to guarantee accurate numerical results. In addition, at this point, we have gained strong confidence in the validity of ES1 and ES3 and in all of the equivalence statements.

_{0}and M vary with R

_{0}for the two series when r

_{L}is kept at 3 Å. For M, the values obtained for the two series overlap (upper black and blue circles); for X

_{0}, the values (lower black and blue circles) obtained for the two series are so close that the corresponding diffusion profiles (see panel b) are visually indistinguishable.

#### 3.4. The HSJE

_{0}and M for the distribution series with b = 2R

_{0}, the series that guarantees highest numerical accuracy by leading to the largest ∆R

_{eff}= L − R differences or DI(X) amplitudes. We obtained smooth curves that could be fitted to second-degree polynomial functions. Data points and fitting curves coincided (Figure 7). The functions and coefficients are given in Table 2.

_{0}= 9–15 Å and r

_{L}= 1.5–5 Å (see the legend of Figure 7) in Figure 8. For clarity, we have decomposed the HSJE into its simple constituents, equations (I) to (VIII). The HSJE written as a single equation is shown in Supporting Information, Chapter 7e.

## 4. Discussion

_{L}. Experimental values of the effective distance can be obtained from time-resolved fluorescence measurements but equally well from more widely available steady-state fluorescence spectroscopy. The HSJE is certainly not trivial but is still simple enough that multivariate or global analysis can be performed in commonly available programs such as Microsoft Excel.

_{D}, the donor lifetime, and decreases with the square root of the Förster radius, R

_{0}(Equation (18)). All of these variations have been employed in earlier attempts at a global analysis but without knowing which range of the diffusion influence was actually covered [3,7,9,10,14]. In this work, we obtained certainty that our previous variation [14] of the diffusion coefficient by solvent viscosity and variation of the donor lifetime by adding the viscogen ethylene glycol and by going from the donor FTrp to the donor NAla covered a DI range of 8 to 80% (comp. Supporting Information, Chapter 8c). In previous works [12,14], we also investigated and discussed the role of the donor quantum yield, and we continue this discussion in Supporting Information, Chapter 9, because the donor quantum yield can be made part of the HSJE as shown in Equations (S31)–(S33).

_{0}-values and under which circumstances are we not close to this limit. Up to which R

_{0}-values should diffusion be included into the analysis, and up to which values should the HSJE be extended?

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**DFRET: Diffusion-enhanced Förster resonance energy transfer. After excitation, the excited donor, D, can either become deactivated by emitting fluorescence (k

_{rad}) or by being quenched, for instance, by iodine ions (k

_{nrad}) or by transferring its excitation energy to the acceptor, A (k

_{FRET}). FRET can take place at every D-A distance but is more likely to happen at shorter distances, which is why it is enhanced by D-A diffusion. The pertinent equations (I–IV) are explained in the main text.

**Figure 2.**(

**a**) An exemplary probability distance distribution of the donor–acceptor distance in a linear (bio)polymer (

**b**) The diffusion profile or R

_{eff}(D

^{1/2}) profile: Solving the HSE under variation of the diffusion coefficient results in values of the effective distance that range from L to R (dashed lines). The effective distance plotted against D

^{1/2}approaches R

_{eff}= L when the extent of diffusional motion approaches zero and approaches R

_{eff}= R when the extent of diffusional motion approaches infinity. (

**c**) The diffusion-influence profile or DI(D

^{1/2}) profile is obtained when the diffusion profile shown in (

**b**) is normalized with L and R according to DI = (R

_{eff}− L)/(R − L) (Equation (11)). The diffusion influence, DI, can adapt values between 0 (0%) and 1 (100%). (

**d**) The diffusion-influence profile, the DI(J

^{1/2}) or DI(X) profile: The diffusion influence plotted against the square root of the augmented diffusion coefficient, X = J

^{1/2}(see Equation (12)).

**Figure 3.**(

**a**) Three different 3-D Gaussian distance distributions (black, red, blue). (

**b**) The corresponding diffusion profiles (black, red, blue) with the effective distance plotted against the square root of the diffusion coefficient. The donor lifetime, the Förster radius, and the left integration limit were kept constant (τ

_{D}= 100 ns, R

_{0}= 10 Å, r

_{L}= 2.5 Å) (

**c**) After normalization (Equation (11)), the three profiles became identical.

**Figure 4.**(

**a**) A 3-D Gaussian distance distribution (

**b**) Four diffusion profiles obtained with (

**a**) four different donor lifetime constants, τ

_{D}, of 100 ns (red), of 30 ns (blue), of 10 ns (green), and of 1 ns (black). The Förster radius (R

_{0}= 15 Å) and the left integration limit (r

_{L}= 3 Å) were held constant. (

**c**) Diffusion-influence profiles after normalization. (

**d**) The DI-profiles coincide when the DI values are plotted against the square root of the product of diffusion coefficient and donor lifetime.

**Figure 5.**(

**a**) Three ideal-chain distance distributions (Equation (18)) with b = 18 Å, r

_{L}= 3 Å (black curve); b = 24 Å, r

_{L}= 4 Å (red curve); and b = 30 Å, r

_{L}= 5 Å (blue curve). (

**b**) The corresponding diffusion profiles obtained with b = 18 Å, r

_{L}= 3 Å, R

_{0}= 9 Å (black curve); with b = 24 Å, r

_{L}= 4 Å, R

_{0}= 12 Å (red curve); and with b = 30 Å, r

_{L}= 5 Å, R

_{0}= 15 Å (blue curve). Thus, for all three evaluated distributions (black, red, blue), the ratio R

_{0}/r

_{L}equaled 3. (

**c**) The corresponding diffusion-influence profiles with DI plotted against X (X = J

^{1/2}). The three profiles merge into one.

**Figure 6.**The HSE was solved for two different series of ideal-chain distributions (Equation (18)). R

_{0}was varied from 9 Å to 15 Å, and r

_{L}was varied from 1.5 Å to 5 Å. The donor lifetime was held constant (τ

_{D}= 100 ns). In one series of distributions, the constant b was chosen as b = 1.5⋅R

_{0}ranging from 13.5 Å to 22.5 Å, in the other series as b = 2⋅R

_{0}ranging from 18 Å to 30 Å. (

**a**) Exemplary distributions for the two series with r

_{L}= 3 Å, R

_{0}= 10 Å and either b = 1.5⋅R

_{0}= 15 Å (blue curve) or b = 2⋅R

_{0}= 20 Å (black curve). (

**b**) The DI(X) profiles overlap for both distributions shown in (

**a**). (

**c**) All DI(X) profiles were analyzed by using Equation (13).

**Figure 7.**The results for X

_{0}and M (solid circles in panels (

**a**,

**b**) obtained for the range of R

_{0}= 9–15 Å, r

_{L}= 1.5–5 Å with b = 2R

_{0}, and R

_{0}/r

_{L}> 3) were fitted to second-degree polynomial functions (solid lines in panels (

**a**,

**b**) given in Table 2).

**Figure 8.**The Haas-Steinberg–Jacob equation (HSJE) is a closed analytical equation decomposed here into simple equations, I to VIII, for clarity. With the coefficients shown here (a

_{0}to b

_{3}), it is valid for R

_{0}9–15 Å and r

_{L}1.5–5 Å.

R_{0}/r_{L} | R_{0} | r_{L} | X_{0} | M |
---|---|---|---|---|

3 | 9 | 3 | 0.8935 | 1.7823 |

3 | 12 | 4 | 0.8929 | 1.7876 |

3 | 15 | 5 | 0.8933 | 1.7861 |

4 | 10 | 2.5 | 1.1722 | 1.6513 |

4 | 12 | 3 | 1.1719 | 1.6532 |

4 | 14 | 3.5 | 1.1724 | 1.6494 |

5 | 10 | 2 | 1.4259 | 1.5387 |

5 | 15 | 3 | 1.4256 | 1.5391 |

6 | 9 | 1.5 | 1.6591 | 1.4481 |

6 | 12 | 2 | 1.6592 | 1.4483 |

6 | 15 | 2.5 | 1.6587 | 1.4483 |

_{0}.

X_{0} = a_{0} + a_{1} × (R_{0}/r_{L}) + a_{2} × (R_{0}/r_{L})^{2}; M = b_{0} + b_{1} × (R_{0}/r_{L}) + b_{2} × (R_{0}/r_{L})^{2} | ||||
---|---|---|---|---|

n | X_{0} = | a_{n}-value | M = | b_{n}-value |

0 | a_{0} | −0.023191 | b_{0} | 2.262606 |

1 | a_{1} × (R_{0}/r_{L}) | 0.333543 | b_{1} × (R_{0}/r_{L}) | −0.185375 |

2 | a_{2} × (R_{0}/r_{L})^{2} | −0.008843 | b_{2} × (R_{0}/r_{L})^{2} | 0.008271 |

_{0}, a

_{1}, a

_{2}, and b

_{0}, b

_{1}, b

_{2}are valid for the conditions R

_{0}= 9–15 Å, r

_{L}= 1.5–5 Å, R

_{0}/r

_{L}> 3.

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## Share and Cite

**MDPI and ACS Style**

Jacob, M.H.; D’Souza, R.N.; Lazar, A.I.; Nau, W.M.
Diffusion-Enhanced Förster Resonance Energy Transfer in Flexible Peptides: From the Haas-Steinberg Partial Differential Equation to a Closed Analytical Expression. *Polymers* **2023**, *15*, 705.
https://doi.org/10.3390/polym15030705

**AMA Style**

Jacob MH, D’Souza RN, Lazar AI, Nau WM.
Diffusion-Enhanced Förster Resonance Energy Transfer in Flexible Peptides: From the Haas-Steinberg Partial Differential Equation to a Closed Analytical Expression. *Polymers*. 2023; 15(3):705.
https://doi.org/10.3390/polym15030705

**Chicago/Turabian Style**

Jacob, Maik H., Roy N. D’Souza, Alexandra I. Lazar, and Werner M. Nau.
2023. "Diffusion-Enhanced Förster Resonance Energy Transfer in Flexible Peptides: From the Haas-Steinberg Partial Differential Equation to a Closed Analytical Expression" *Polymers* 15, no. 3: 705.
https://doi.org/10.3390/polym15030705