Effects of diffusion on energy transfer in solution using a microsecond decay time rhenium metal–ligand complex as the donor

Abstract

We used resonance energy transfer and frequency-domain fluorometry to measure slow donor to acceptor diffusion in viscous media. The frequency-domain RET data were analyzed using a new numerical algorithm for predicting the donor intensity decay in the presence of diffusion occurring within the donor decay time. By the use of a rhenium metal–ligand complex as a microsecond decay time donor we were able to measure mutual donor-to-acceptor diffusion coefficients as low as 2 × 10⁻⁸ cm²/s. The availability of microsecond decay time luminophores and appropriate theory suggests the use of diffusion-enhanced energy transfer for measurement of diffusive processes and structural dynamics in biological systems.

1. Introduction

Fluorescence resonance energy transfer (RET) is widely used to study the structure of biomolecules [1–3]. Most fluorophores decay on the ns timescale during which there is little time for donor-to-acceptor motions. It is well known that diffusive motions during the excited state lifetime of the donor increase the efficiency of energy transfer [4,5]. However, the diffusion coefficients must be moderately large to affect the RET efficiency of donors with nanosecond lifetimes. The use of RET to measure structural dynamics has remained elusive due to the lack of long-lifetime fluorophores.

Over the past five years there has been considerable progress in the development of long-lifetime fluorophores. This laboratory [6–8] and others [9,10] focused on developing transition-metal complexes (MLCs) of Ru, Re or Os which contain at least one diimine ligand. The class of molecules has been extensively studied because of their interesting photophysics and because of their possible use in solar energy conversion [11,12]. These MLCs display lifetimes ranging from 10 ns to 10 μs. Those MLCs which display microsecond decay times can potentially be used to study microsecond motions in proteins, membranes and nucleic acids [13].

In the present Letter we initiate the use of MLCs to study microsecond dynamics. We use a Re MLC which displays decay times ranging from 2 to 5 μs. Using this MLC as a RET donor we were able to measure donor-to-acceptor diffusion coefficients as low as 2 × 10⁻⁸ cm²/s, which would not be measurable using an donor with a nanosecond decay time. Such diffusion coefficients are comparable to those expected for lateral diffusion in membranes or for domain motions in proteins.

2. Materials and methods

The N-hydroxy succinimide ester of Texas Red was obtained from Molecular Probes, Eugene, OR. We believe the NHS ester remained intact in the propylene glycol (PG) solutions, and we refer to their NHS ester as simply Texas Red (TR). Re(CO)₅Cl, 5,6-dimehyl-1,10-phenanthroline (5,6-dmphen), isonicotinic acid (py-COOH), silver perchlorate, N-hydroxysuccinimde (NHS), dicyclohexylcarboxydiimide (DCC), toluene, dichloromethane, acetonitrile, and methanol were obtained from Aldrich and were used as received without further purification. Re(5,6-dmphen)(CO)₃(py-COOH)ClO₄ was synthesized according to the literature procedure [14].

2.1. Synthesis of Re(5,6-dmphen)(CO)₃(py-CONH-(CH₂)₁₂NHCOCH₃ · ClO₄

To 0.185 g of Re(5,6-dmphen)(CO)₃(py-COOH) ClO₄ in 50 ml of dichloromethane, 0.04 g of N-hydroxysuccinimide, and 0.06 g of dicyclohexylcarboxydiimide were added and stirred for 24 h. The resultant solution of the Re-NHS ester was filtered and 0.30 g of 1,12-diaminododecane was added and further stirred for 24 h. The solution was filtered and vacuum dried to yield Re(5,6-dmphen)(CO)₃(py-CONH(CH₂)₁₂ClO₄. The N-acetyl derivative was prepared by known procedures.

Frequency-domain intensity decays were measured as described previously [15]. The intensity decays were initially fit to the multi-exponential model when the intensity decay is given by

$$I(t)=\sum _i{\alpha }_i\exp \left(-t/{\tau }_i\right),$$ (1)

where α_i are the time-zero amplitudes due to the components with decay times τ_i, and Σα_i = 1.0. The light source was the 325 nm output of a HeCd laser amplitude modulated using a Pockel’s cell. The donor emission was selected using an emission filter transmitting from 500 to 540 nm and magic angle polarizer conditions. The data were analyzed by non-linear least squares. The goodness-of-fit parameter ${\chi }_{\text{R}}^2$ was calculated using

$${\chi }_R^2=\frac{1}{\nu }\sum _{\omega }{\left[\frac{{\phi }_{\omega }-{\phi }_{\text{c}\omega }}{\mathrm{\delta }\phi }\right]}^2+\frac{1}{\nu }\sum _{\omega }{\left[\frac{m_{\omega }-m_{\text{c}\omega }}{\mathrm{\delta }m}\right]}^2,$$ (2)

where v is the number of degrees of freedom and ω indicates the light modulation frequency. The value of v is given by the number of measurements, which is typically twice the number of frequencies minus the number of variable parameters. The subscript c is used to indicate calculated values for assumed values of α_i and τ_i, and δϕ and δm are the uncertainties in the phase (ϕ_ω) and modulation (m_ω) values, respectively.

3. Theory

The use of RET for distance measurements on macromolecules is relatively straightforward because of a single acceptor at a fixed distance [1,2]. Hence all of the D–A pairs are present in a single unique conformation. The theory becomes considerably complex for donors and acceptors in solution because of the presence of a statistical ensemble of D–A distances. For instance, for a random solution of donors and acceptors in three dimensions, and no diffusion during the excited state lifetime, the donor intensity decay is given by

$$I_{\text{DA}}(t)=I_{\text{D}}^0\exp \left[-\frac{t}{{\tau }_{\text{D}}}-2\gamma {\left(\frac{t}{{\tau }_{\text{D}}}\right)}^{1/2}\right],$$ (3)

where τ_D is the donor decay in the absence of acceptors and γ is a constant related to the Förster distance (R₀). The value of γ is given by: $\gamma \approx 3.7122C_{\mathrm{A}}^0{R}_0^3$, where $C_{\mathrm{A}}^0$ is the acceptor concentration. Now suppose one allows for donor-to-acceptor diffusion during the excited state lifetime. No analytical expression is known to describe the donor decay in the presence of D-to-A diffusion. In these cases the donor decay is described by expressions which approximate the intensity decay [16,17].

To circumvent this problem we developed a numerical solution to the diffusion equation. We assume the intensity decay of the donor in the absence of acceptor is a multi-exponential

$$I_{\text{D}}(t)=I_{\text{D}}^0\sum _i{\alpha }_i\exp \left(-t/{\tau }_{\text{D}i}\right).$$ (4)

The values of α_i are the time-zero amplitudes of each component which are assumed to remain the same in the presence of acceptors, and τ_Di are the donor decay times in the absence of acceptors.

In the presence acceptor the intensity decay becomes

$$\begin{gathered} I_{\text{DA}}(t)=I_{\text{D}}^0\sum _i{\alpha }_i\exp \left[-t/{\tau }_{\text{D}i}-C_{\text{A}}^0{W}_i(t)\right] \\ =\sum _i{I}_{\text{DA}i}(t), \end{gathered}$$ (5)

where $C_{\text{A}}^0$ is the bulk concentration of the acceptor (number of acceptor molecules per cm³ ) and

$$W_i(t)={\int }_0^t{k}_i\left(t^{\prime }\right)\text{d}{t}^{\prime }.$$ (6)

In Eq. (6) k_i(t) denotes the time-dependent second-order transfer rate which within this three-dimensional energy transfer model is defined as

$$k_i(t)=4\pi {\int }_{r_{\min }}^{\infty }{r}^2{k}_{\text{DA}i}(r)y_i(r,t)\text{d}r,$$ (7)

where r_min denotes the distance of donor-acceptor closest approach. It is assumed that the Förster radius R₀ has the same value for all intensity components, i.e., that the transfer rates k_{DA i}(r) have the form

$$k_{\text{DA}i}(r)=\frac{1}{{\tau }_{\text{D}i}}{\left(\frac{R_0}{r}\right)}^6.$$ (8)

Functions y_i(r, t) have a meaning of ratios of the mean concentration C_{A i}(r, t) of acceptor molecules at the distance r from the excited donor of the ith type to the bulk concentration of the acceptor $C_{\text{A}}^0$. These ratios satisfy the diffusion equation

$$\frac{\partial y_i(r,t)}{\partial t}=D{\nabla }^2{y}_i(r,t)-k_{\text{DA}i}(r)y_i(r,t),$$ (9)

where D = D_D + D_A is the sum of the diffusion coefficients of the donor and acceptor, respectively. The initial condition of Eq. (9) is

$$y_i(r,t=0)=1$$ (10)

and the inner and outer boundary conditions are

$${\left[\frac{\partial y_i(r,t)}{\partial r}\right]}_{r=r_{\min }}=0,$$ (11)

and

$$y_i(r\to \infty ,t)=1,$$ (12)

respectively. An analytical solution of Eq. (9) is not known, so numerical methods were applied [18]. In order to simplify calculations Eq. (9) is transformed to the Laplace space. The resulting linear differential equation

$$\begin{gathered} \frac{{\text{d}}^2{y}_i(r,p)}{\text{d}{r}^2}+\frac{2}{r}\frac{\text{d}{y}_i(r,p)}{\text{d}r} \\ -\frac{1}{D}\left[p+k_{\text{DA}i}(r)\right]y_i(r,p)=-\frac{1}{D} \end{gathered}$$ (13)

is solved using the relaxation method. For given set of parameters p, D, R₀, r_min, and τ_i the solution of the equation consists of 200 values of the function y_i(r, p) calculated at 200 points r_k equally spaced in the interval (r_min, r_ε). The value of r_ε, the upper limit of the interval, is estimated every time as fulfilling the condition

$$p{\left[\frac{\text{d}{y}_i(r,p)}{\text{d}r}\right]}_{r=r_{\epsilon }}=\epsilon ,$$ (14)

where ε is a small number of the order of 10⁻⁴. This procedure minimizes the length of the calculation interval on the r-axis, while simultaneously taking the boundary condition (12) into account. It is assumed in further calculations that for r > r_ε the functions y_i(r, p) are independent of r and equal to 1/p. After that, the quantities W_i(p) are calculated using the relation

$$\begin{gathered} W_i(p)=\frac{k_i(p)}{p}=\frac{4\pi }{p}[{\int }_{r_{\min }}^{r_{\epsilon }}{r}^2{k}_{\text{DA}i}(r)y_i(r,p)\text{d}r \\ +{\int }_{r_{\epsilon }}^{\infty }{r}^2{k}_{\text{DA}i}(r)\text{d}r]. \end{gathered}$$ (15)

Finally, the W_i(p) values are inverted to the time space using the Stehfest procedure [19,20].

For least-squares analysis this formalism was used to calculate the donor decays for assumed values of D and the acceptor concentration. These predicted intensity decays were used to calculate the sine and cosine transform at a circular modulation frequency ω using

$$N_{\omega }=\frac{\sum _i{\int }_0^{\infty }{I}_{\text{DA}i}(t)\sin (\omega t)\text{d}t}{\sum _i{\int }_0^{\infty }{I}_{\text{DA}i}(t)\text{d}t},$$ (16)

$$D_{\omega }=\frac{\sum _i{\int }_0^{\infty }{I}_{\text{DA}i}(t)\cos (\omega t)\text{d}t}{\sum _i{\int }_0^{\infty }{I}_{\text{DA}i}(t)\text{d}t}.$$ (17)

These transforms are then used to calculate the phase (ϕ_cω) and modulation (m_cω) using

$$\tan {\phi }_{\text{c}\omega }=\frac{N_{\omega }}{D_{\omega }},$$ (18)

$$m_{\text{c}\omega }={\left(N_{\omega }^2+D_{\omega }^2\right)}^{1/2}.$$ (19)

The calculated and measured intensity decays were matched by minimization of ${\chi }_R^2$ (Eq. (2)). The intensity decays were also used to calculate a predicted steady-state donor intensity (F_DA) relative to that in the absence of acceptors (F_D) using

$$\frac{F_{\text{DA}}}{F_{\text{D}}}=\frac{{\int }_0^{\infty }{I}_{\text{DA}}(t)\text{d}t}{{\int }_0^{\infty }{I}_{\text{D}}(t)\text{d}t}.$$ (20)

The transfer efficiency (E) is given by

$$E=1-\frac{F_{\text{DA}}}{F_{\text{D}}}.$$ (21)

4. Results

Chemical structures of the ReMLC donor and the Texas Red acceptor are shown in Scheme 1. The dmphen and carboxypyridine ligands were chosen to provide a good quantum yield and long decay time. Texas Red was chosen as the acceptor because of the overlap of its absorption with emission from the ReMLC (Fig. 1). Also, the Texas Red emission is well shifted from the MLC emission making it possible to observe only the donor emission when using an appropriate emission filter. The quantum yield of the ReMLC propylene glycol was found to be 0.054 and 0.19 at 20 and −20°C, respectively, as measured relative to 3-aminofluoranthene in dimethyl-sulfoxide [21]. Using these values we calculated Förster distances of 35.5 and 43.7 Å at 20 and −20°C, respectively.

Scheme 1.

Chemical structure of the Re-MLC donor and the Texas Red acceptor.

Fig. 1.

Spectral overlap of the Re-MLC donor emission (—) with the absorption of the Texas Red acceptor (— — —). The Förster distance at 20°C in propylene glycol is estimated to be 35.5 Å. The long dashed line shows the transmission profile of the emission filter used for the intensity decay measurements. The scheme on the right shows the geometry used to minimize the inner filter effects using a 1 cm by 0.5 mm dye laser cuvette. Ex, excitation; Em, emission.

Prior to examining experimental data it is valuable to understand the effects of diffusion on the RET efficiency. Fig. 2 shows the transfer efficiency calculated for a range of acceptor concentrations and mutual diffusion coefficients. In the slow diffusion limit the RET efficiency becomes constant at the value which depends on the acceptor concentration. These efficiencies are equal to those calculated using the time-integrated form of Eq. (3) [22]

$$\frac{F_{\text{DA}}}{F_{\text{D}}}=1-\sqrt{\pi }\gamma \exp \left({\gamma }^2\right)[1-\mathrm{erf}(\gamma )],$$ (22)

where

$$\mathrm{erf}(\gamma )=\frac{2}{\sqrt{\pi }}{\int }_0^{\gamma }\exp \left(-x^2\right)\text{d}x.$$ (23)

This agreement demonstrates the validity of our algorithm in the no-diffusion limit (Table 1).

Fig. 2.

Effect of diffusion on the efficiency of RET in three dimensions. For these simulations the parameter values were τ_D = 5 μs, R₀ = 40 Å, r_min = 5 Å. C_A is the molar acceptor concentration.

Table 1.

Comparison of RET transfer efficiencies calculated with our algorithm and analytical expressions for high and low rates of diffusion

[Acceptor]	No diffusion		Fast diffusion
	static theorya	our algorithm	rapid diffusive limitb	our algorithm
10−5	0.003	0.003	0.461	0.437
10−4	0.025	0.025	0.895	0.886
10−3	0.217	0.220	0.988	0.987
5 × 10−3	0.659	0.664	0.998	0.997
10−2	0.845	0.848	0.999	0.999
10−1	0.998	0.998	1.000	1.000

^aCalculated using Eqs. (22) and (23) [22].

^bCalculated using Eqs. (25) and (24) from Ref. [23] and the parameter values given in Fig. 2.

In the limit of rapid diffusion the transfer efficiency becomes independent of acceptor concentration (Table 1). For spherical donors and acceptors the transfer rate k_T in the rapid diffusion limit is given by

$$k_{\text{T}}=C_{\text{A}}^0{\int }_{r_{\text{min}}}^{\infty }\frac{1}{{\tau }_{\text{D}}}{\left(\frac{R_0}{r}\right)}^{6}4\pi r^2\text{d}r=\frac{4\pi C_{\text{A}}^0{R}_0^6}{3{\tau }_{\text{D}}{r}_{\text{min}}^3},$$ (24)

where $C_{\text{A}}^0$ is the density of acceptors (molecules/ų) and r_min is the distance of closest approach. The diffusion-limited transfer rate is thus dependent on $r_{\min }^{-3}$ and the acceptor concentration.

The transfer efficiency is given by

$$E=\frac{k_{\text{T}}}{k_{\text{T}}+{\tau }_{\text{D}}^{-1}}.$$ (25)

The limiting values of the transfer efficiency using our algorithm are equal to those calculated using the rapid diffusion limit [22,23]. Additionally, the donor intensity decays calculated with our algorithm agree closely with the known numerical approximation (not shown) [16].

Interesting behavior was found for intermediate diffusion coefficients and acceptor concentrations. For diffusion coefficients near 10⁻⁸ to 10⁻⁷ cm²/s the transfer efficiency is strongly dependent on D and the acceptor concentration. This dependence is strongest near mM acceptor concentrations. In this region one can expect the RET efficiency and the time-resolved decays for microsecond decay time donors to contain information on moderately slow diffusion in solution.

We measured the frequency domain intensity decay of the ReMLC (Scheme 1) in the absence and presence of 8 × 10⁻⁴M Texas Red. The ReMLC decay was moderately heterogeneous (Table 2). The mean decay time in propylene glycol was dependent on temperature and increased from 2.15 μs at 20°C to 7.45 μs at −20°C. One may question the higher values of ${\chi }_R^2$ at the lower temperatures (Table 2). This effect is due to the decreased performance of our RF amplifiers at the lower frequencies, resulting in more noise in the data. The presence of acceptor shifted the frequency response to high frequencies, indicating a decreased donor lifetime (Fig. 3). In order to visualize the contribution of diffusion to the decreased lifetime we used our theory with D = 0 to predict the frequency response expected for the known R₀ and acceptor concentration (Fig. 3, — — —). These calculated curves for 8 × 10⁻⁴ M acceptor are shifted towards higher frequencies, but to an extent less than the experimental data. The hatched areas between these curves and the experimental data (– • –) represent the extent to which the frequency responses are further modified by diffusion. At 20°C most of the observed shift is due to diffusion. Surprisingly, diffusion still makes a dominant contribution to the RET efficiency even at −20°C where diffusion is slow. This sensitivity to slow diffusion is due in part to the 2.4-fold increase in the decay time of the ReMLC at −20°C.

Table 2.

Intensity decay analysis of Re-MLC donor in absence and presence of TR acceptor

T (°C)	${\mathit{C}}_{\mathbf{A}}^{\mathbf{0}}(\mathit{M})$	Multi-exponential analysis					Energy transfer analysis
		τ1 (ns)	τ2 (ns)	α1	$\overline{\mathit{\tau }}$a (ns)	${\mathit{\chi }}_{\mathit{R}}^{\mathbf{2}}$	R0 (Å)	D (cm2/s)	${\mathit{\chi }}_{\mathit{R}}^{\mathbf{2}}$
20	0	173	2202	0.247	2150	3.1	–	–	–
	8 × 10−4	224	1304	0.287	1233	1.0	< 35.5 >	3.34 × 10−7	1.3
							39.3	2.52 × 10−7	1.2
−20	0	1882	7625	0.115	7446	2.9	–	–	–
	8 × 10−4	1177	5326	0.217	5086	3.1	< 43.7 >	2.40 × 10−8	2.8
							47.0	1.62 × 10−8	2.5

^a$\overline{\tau }={\sum }_i{f}_i{\tau }_i$, $f_i=\frac{{\alpha }_i{\tau }_i}{\sum {\alpha }_j{\tau }_j}$.

Fig. 3.

Intensity decays of the Re-MLC donor (●) in the presence of 8×10⁻⁴ M Texas Red acceptor in propylene glycol at 20°C (top) and −20°C (bottom). The dotted lines are the donor intensity decays in the absence of acceptor. The dashed lines are the predicted frequency responses with no diffusion. The shaded areas show the effect of diffusion on the donor intensity decay.

We analyzed the FD intensity decays in terms of RET in three dimensions. The data were analyzed two ways, with R₀ fixed at the known value and with both R₀ and D as variable parameters (Table 2). Essentially equivalent fits were obtained, with recovered diffusion coefficients near 3 × 10⁻⁷ and 2 × 10⁻⁸ cm²/s in propylene glycol at 20 and −20°C, respectively. Examination of Table 2 reveals that somewhat different values for the diffusion coefficient with R₀ held fixed or used as a variable parameter. Even though the diffusion coefficients are different, the values of ${\chi }_R^2$ are essentially the same. This occurs because the values of R₀ and D are correlated parameters in the fitting procedure. While least-squares fitting programs typically return the uncertainties in the recovered parameters, these uncertainties are usually calculated assuming the parameters are not correlated. The confidence intervals for the parameter can be substantially larger if the parameters are correlated [23,24].

One way to accurately determine the confidence interval is from the ${\chi }_R^2$ surfaces (Fig. 4). These surfaces show how the value of ${\chi }_R^2$ depends on the parameter value. For the diffusion coefficient steeper ${\chi }_R^2$ surfaces were found when the R₀ value was constant (— — —) than when R₀ was variable (— —). In either case the surfaces show good resolution of diffusion coefficients as low as 1.6 × 10⁻⁸ cm²/s. The maximum range of diffusion coefficients is found from the intersection of the ${\chi }_R^2$ surface with the F_χ value appropriate for the experiment. This value is calculated from [25]

$$F_{\chi }=\frac{{\chi }_R^2(D)}{{\chi }_R^2\left(D_{\min }\right)}=1+\frac{1}{\nu }F(p,\nu ,P),$$ (26)

where the ${\chi }_R^2$ values are calculated at the minimum (D_min) and other values of D, p is the number of parameters, and v is the number of degrees of freedom. For one parameter (p = 1) and one standard deviation (P = 0.32), the F-statistic appropriate for this experiment is F(1, 20, 0.32) ≈ 1.05.

Fig. 4.

${\chi }_R^2$ surfaces for the mutual D-to-A diffusion coefficient with R₀ held constant (— — —) or allowed to vary (— —) while minimizing ${\chi }_R^2$. The y-axis ${\chi }_{R\text{N}}^2$ is the ratio or ${\chi }_R^2$ values defined in Eq. (26).

In order to have a conservative estimate of our precision we used a value of F_χ= 1.1. Assuming the worse case situation where R₀ is unknown this value of F_χ yields a range of is D = (1.15–2.45) × 10⁻⁸ cm²/s at −20°C. If R₀ is known, which is typically true in a RET experiment, then the range of D values consistent with the data at −20°C decrease to 2.15 to 2.80 × 10⁻⁸ cm²/s.

5. Discussion

The ability to measure slow diffusive transport can be widely applicable in biochemistry and biophysics. As examples we note that lipid probes with microsecond decay times can be used to measure lateral diffusion of lipids in membranes [26] and is likely to be useful in measuring domain motions in proteins [27]. Additionally, one can imagine RET on the microsecond timescale to be used to estimate the overall viscosity of intercellular environments. Because of the availability of microsecond decay time probes, fluorescence spectroscopy is no longer trapped on the nanosecond timescale.