and cover glasses for Quartz Crystal Microbalance with Dissipation (QCM-D), Surface ... response with final frequency shift of â27±1.8 Hz and dissipation shift of ...
Supplementary Information
Detachment of membrane bound virions by competitive ligand-binding induced receptor depletion Nagma Parveen1, Stephan Block1,†, Vladimir P. Zhdanov1,2, Gustaf Rydell3, Fredrik Höök1
1 2 3
Department of Physics, Chalmers University of Technology, Gothenburg, Sweden
Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk, Russia
Department of Infectious Diseases, Sahlgrenska Academy, University of Gothenburg, Gothenburg, Sweden
†
Department of Chemistry and Biochemistry, Freie Universität Berlin, Berlin, Germany
TABLE OF CONTENTS Section S1 : Materials and preparation details
2
Section S2 : Supportive experiments with QCM-D and SPR
2
Section S3 : TIRFM assay and control experiments
6
Section S4 : Theoretical background
8
S1. Materials and preparation details A. Materials. 1-Palmitoyl-2-Oleoyl-sn-Glycero-3-Phosphocholine (POPC), Monosialoganglioside (GM1) from bovine brain and Lissamine Rhodamine B 1,2Dihexadecanoyl-sn-Glycero-3-Phosphoethanolamine (Rhodamine-DHPE) were purchased from Avanti Polar Lipids (USA), Sigma-Aldrich (Germany) and Invitrogen (USA), respectively. Recombinant VLP of simian virus 40 (SV40) and recombinant cholera toxin B subunit (rCTB) were purchased from ABCAM (UK) and Sigma Aldrich (Germany), respectively. B. Vesicle preparation. POPC, GM1 and Rhodamine-DHPE lipids dissolved in chloroform or methanol were pipetted into a round bottom flask. The lipid mixture was first dried under a gentle N2 stream and then dried in vacuum (minimum 2 hrs) to obtain a lipid film. This dried film was hydrated in TRIS buffer (20 mM TRIS, 100 mM NaCl, pH 7.5) with help of a vortex. Lipid vesicles were prepared by extruding (15 times) this hydrated lipid suspension with polycarbonate filter (Whatman, UK) of 30 nm pore size. POPC, GM1 and Rhodamine-DHPE lipids were mixed in desired weight ratio to control the vesicle composition. The size distribution of the extruded vesicles was 90 ± 40 nm determined with a Nanoparticle Tracking Analysis (NTA) device (Malvern, UK). C. Supported lipid bilayer (SLB) formation. SLBs were prepared by spreading the extruded lipid vesicles on silica-coated quartz crystals, silica-coated gold plasmon sensors and cover glasses for Quartz Crystal Microbalance with Dissipation (QCM-D), Surface Plasmon Resonance (SPR) and Total Internal Reflection Fluorescence (TIRF) experiments, respectively. GM1 content in the SLB was regulated mixing two batches of pure POPC and of POPC+5 wt% GM1 vesicles in desired volume ratio. Two-phase change in QCM-D response with final frequency shift of −27±1.8 Hz and dissipation shift of (0.2±0.15)×10-6 indicates a good quality SLB formation. SLB formation on SPR sensors and cover glasses was confirmed by measuring the diffusion coefficient of tracer fluorescent RhodamineDHPE lipids with florescence recovery after photobleaching (FRAP). Further, the GM1 distribution in SLB was checked using fluorescently labeled Bodipy-GM1 using TIRFM. The Bodipy-GM1 content in the SLB was controlled by varying the mixing ratio of suspensions containing vesicles made of POPC and POPC+5 wt% Bodipy-GM1. The spatial distribution of the emission signal from incorporated Bodipy-GM1 in the SLB was homogenous and the relative signal intensity was proportional to the expected BodipyGM1 content based on the mixing procedure. The diffusivity of Bodipy-GM1 in the SLB was determined with FRAP to be 1.2±0.3 µm2/s, which is similar to that of POPC. S2. Supportive experiments with QCM-D and SPR A. QCM-D assay. SiO2-coated quartz crystal sensors were used as substrates for QCM-D (QSense E4 instrument; QSense, Göteborg, Sweden) experiments. They were immersed in 2 wt% SDS solution overnight and then washed thoroughly with ultrapure water (MilliQ) followed by UV-Ozone treatment for 30 minutes.
2
-2 -4
3
2
-6 1
-8
-10
0 50
75
100
125
-dΔf/dt (Hz/min)
Frequency, Δf (Hz)
2% 1% 0.5% 0.25% 0.1%
0.20
4
Dissipation, ΔD (10−6)
a)
0
b)
0.15 0.10 0.05 0.00 0.0
150
Time (min)
0.5 1.0 1.5 wt% of GM1 in SLB
2.0
Figure S1. rCTB binding to GM1 in SLB. a) QCM-D time trace (5th overtone) of rCTB binding on SLBs with different GM1 concentrations. Binding was done from 3.45 nM rCTB solution at flow rate of 50 µL/min. b) Binding rate of rCTB in terms of –dΔf/dt versus GM1 concentration in SLB. According to Sauerbrey relation [1], frequency response of −10 Hz for POPC+2wt% GM1 corresponds a wet surface coverage of 1.5x10−2 nm-2 rCTB. Both rCTB and coupled water contribute to the estimated QCM-D surface coverage. B. SPR assay. The corresponding experiments were performed with a multiparametric SPR (MP-SPR) Navi 220A (BioNavis, Finland). The SPR sensors were silicacoated gold plasmon surfaces on glass substrates and were cleaned by first thorough washing with 10 mM SDS solution for at least 10 minutes followed by thorough rinsing with MilliQ water. Then they were dried and kept in UV-Ozone chamber for 10 minutes. Measurements were performed at 22°C and SPR was monitored at wavelengths 670 and 785 nm for a scanning interval between 40 to 78 degree. The dry surface coverage of a film formed by adsorption of particles on a plasmon surface is related to the SPR signal, Δdegree λ , as [2,3] ΔΓ film =
dfilm ⎛ dn ⎞ Sλ ⋅ ⎜ ⎟ ⋅ (1− exp (−dfilm / δ )) ⎝ dC ⎠
⋅ Δdegree λ
(1)
where δ is the penetration depth of the evanescent field of surface plasmon, S is bulk sensitivity parameter, dn/dC is the refractive index increment of the adsorbed particle in suspension and dfilm is the thickness of the adsorbed film. In water, the values of δ are 109 and 154 nm at SPR optical wavelengths of 670 and 785 nm, respectively [4]. The values of S are 109.67 and 91.42 degree/RIU at 670 and 785 nm, respectively, as estimated from a calibration curve by plotting Δ deg vs. the independently determined refractive index of aqueous glycerol solutions at five concentrations. According to the literature reported dn/dC value for proteins is 0.182 mL/g [2,5]. Figure S2 presents the change in the plasmon angle (Δdegree) upon binding of rCTB and SV40 on SLB with 2 wt% GM1. Surface coverage of the adsorbed films was determined by employing in Eq. (1) the above-mentioned values of the SPR parameters and assuming a film thickness of 5 and 40 nm for rCTB and SV40, respectively. The
3
estimated average surface coverages of rCTB and SV40 are 0.6×10−2 and 2×10−4 nm−2, respectively. If the surface coverage determined with SPR is compared with Sauerbrey mass from QCM-D, Δf to Δm conversion factor becomes –5.8 and –18.2 rather than –17.7 ngHz–1cm–2 for rCTB and SV40, respectively and this applies for a dry but not hydrated films. 0.8
a)
1.5
b)
Δdeg
Δdeg
0.6
0.4 buffer buffer
0.2
0.0 20
POPC+ 2wt% GM1 vesicles
670 nm 785 nm
30
40
50
Buffer
0.5
rCTB
POPC+ 2 wt%GM1 vesicles
1.0
60
Buffer
SV40
670 nm 785 nm
0.0
70
60
80
100
120
Time / min
Time / min
Figure S2. Change in the total internal reflection (TIR) angle of surface plasmon resonance (SPR) upon a) rCTB and b) SV40 binding to 2 wt% GM1 in SLB. A SLB was fabricated on SiO2 coated SPR sensor. The respective bindings were done from 17 nM and 0.139 nM rCTB and SV40 solutions at a flow rate of 20 µL/min and 50 µL/min, respectively.
Figure S3. Cryo-TEM image of SV40 VLPs in TRIS buffer at pH 7.5, showing an average SV40 diameter of 40 nm. Sample preparation was done in a CEVS (controlled environment vitrification system), freeze-plunged into liquid ethane (−180°C) and then stored in liquid nitrogen. The Lacey carbon filmed Cu grids were used for the imaging. The microscope is a Philips CM120 cryo-TEM operated at 120 kV and equipped with a cryo-holder (Oxford Instruments CT-3500). Images were recorded with a GIF 100 (Gatan imaging filter). C. Deconvolution method. In our QCM-D experiments the dissipation shift is predominately contributed by SV40 binding, since the dissipation shift upon rCTB binding is close to zero. Moreover, dissipation to frequency response of SV40 is nearly linear in our measured conditions, which is fitted with a linear equation. This fit is used to extract the frequency response of SV40 from the measured dissipation during the rCTB flow in
4
the measured time trace. The extracted frequency trace of SV40 is subtracted from the measured frequency trace to get the frequency trace of rCTB. An example of such deconvolution is given in Figure S4. 4 SV40
2
Dissipation, ΔD (10−6)
Frequency, Δf (Hz)
5
0
0
-2
-5
-4
-10
-6
-15
-8 rCTB
-20 40
50
60
70
80
SV40+rCTB SV40 rCTB
90
100
-10 -12
Time (min)
Figure S4. Deconvolution of QCM-D time traces of SV40 binding followed by rCTB binding with simultaneous SV40 release at 0.25 wt% GM1 in SLB. SV40
a)
-5 -10 -15
4 3 2
-20 -25
1
b)
0.6
1/trelease or kd (min-1)
wt% of GM1 in SLB 2% 0.5%
Dissipation, ΔD (10−6)
Frequency, Δf (Hz)
0
0.4
0.2
rCTB
-30 50
75
100
125
150
175
200
0.0
0
Time (min)
0.0
0.5
1.0
1.5
2.0
wt% of GM1 in SLB
Figure S5. SV40 release kinetics from a constant surface coverage of SV40 at different GM1 content in SLB. a) QCM-D time traces of SV40 binding at two GM1 concentrations with a same final surface coverage followed by rCTB-induced release of the bound SV40. b) Release rate constant vs GM1 concentration, while SV40 surface coverage was almost same at different GM1 wt%. D. Inhibition of SV40 by pre-adsorbed rCTB on SLB. To confirm that the higher binding affinity of rCTB causes SV40 release, we monitored inhibition of SV40 binding in presence of pre-bound rCTB on SLB. Pre-adsorbed rCTB on SLB alters the binding rate of SV40 on SLB (Figure S6a) and the observations can be explained with Eq. (1) in the main 0 text. When nrCTBcrCTB < cGM1 , for example the black curve in Figure S6a, the available GM1
content ( cGM1 ) is appreciably large. Accordingly the subsequent SV40 binding is diffusion controlled resulting nearly constant binding rate of SV40 (Fig. S6b). When 0 nrCTBcrCTB ≥ cGM1 (as most likely PrCTB < 1), for example the red and blue curves in Figure
S6a, cGM1 becomes appreciably low. Thus the binding rate of SV40 decreases. The transition from fully bound state to inhibited state is sharp and similar to the trend in the binding rate of SV40 to free GM1 in SLB shown in Figure 2b in the main text. Here, the
5
length of the plateau in the binding rate of SV40 is proportional to the total GM1 content, 0 cGM1 , in the SLB (Figure S6c). The SV40 inhibition data (Figure S6) complement the
release data shown in the main text and support our reasoning that the higher affinity of rCTB inhibits SV40 binding.
SV40
-5 -10 -15
4 3 2
-20 -25
1 rCTB
-30 50
75
100
125
150
175
200
-dΔf/dt of SV40 (Hz/min)
Frequency, Δf (Hz)
wt% of GM1 in SLB 2% 0.5%
Dissipation, ΔD (10−6)
a)
0
-dΔf/dt of SV40 (Hz/min)
0.8 0.6 0.4 0.2 0.0
0
0
Time (min)
1.50
b)
1.0
1
2
3
4
5
6
-Frequency of rCTB (Hz)
c)
1.25 GM1 in SLB 1 wt% 3.5 wt%
1.00 0.75 0.50 0.25 0.00 0
2
4
6
8
10
12
14
16
Frequency of rCTB (Hz)
Figure S6. a) QCM-D time traces of rCTB and subsequent SV40 binding on a SLB containing 1 wt% GM1. The bindings were done from 0.139 nM SV40 and 3.45 nM rCTB solutions at a flow rate of 50 µL/min for 10 min. Buffer solution was flowed to terminate rCTB binding at the times indicated by arrows. b) Rate of frequency change upon SV40 binding versus the surface coverage of rCTB in terms of its frequency shift at 1 wt% GM1 in SLB. c) Rate of frequency change upon SV40 binding versus the surface coverage of rCTB at 1 and 3.25 wt% GM1 in SLB. S3. TIRFM assay and control experiments A. TIRFM assay. Cover glasses were used for SLB formation and further surface modification in our TIRFM assay. They were cleaned for an hour in 7 times diluted 7X (MPbiomedicals, France) detergent solution. A home-made PDMS stamp consisting of 3 wells (volume 10 µL each) was mounted on the glass substrate [6]. SLBs were formed in individual wells by incubating 10 µL of 0.1 mg/mL vesicle solution for 20 min followed by rinsing with a total of 100 µL of TRIS buffer without drying the surface during the rinsing steps. Afterward, SV40 binding was done by incubating SV40 VLPs for 30 min at a final VLP concentration of 0.00695 nM in the PDMS well. After thorough rinsing with 100 µL of TRIS buffer, POPC vesicles were bound from 100 ng/mL solution to passivate/block
6
potential SLB defects. Then binding of fluorescently labeled POPC+5wt% GM1 vesicles were performed from a final concentration of 100 ng/mL vesicle solution in the PDMS well. After an equilibration time of 30 min, rCTB solution was added at a final concentration of 34.5 nM in the PDMS well. Time-lapse movies were acquired with a time interval of 500 ms for at least 17 minutes (2000 frames). The exposure time for excitation was for 200 ms. Nikon Eclipse Ti-E inverted microscope with a 60× magnification (Numerical aperture (NA)=1.49) oil immersion objective (Nikon Corporation, Tokyo, Japan) was used to acquire time-lapse movies. The microscope was equipped with a mercury lamp (Intensilight C-HGFIE; Nikon Corporation), a TRITC filter cube (Nikon Corporation) and an Andor DU-897 X-3530 EMCCD camera (Andor Technology, Belfast, Northern Ireland). B. SPT. All SPT analysis was done using home-made scripts written in MatLab (MathWorks, Natick, MA). SPT was implemented using local nearest-neighbour linking [7]. Diffusion coefficients were calculated using the internal averaging procedure on a moving window of the data points [8].
a)
b)
Figure S7. Negative control experiments with TIRFM. Binding of fluorescently labeled POPC+5 wt% GM1 vesicles to a) SLB with 0.25 wt% GM1 and b) rCTB bound on a SLB with 0.25 wt% GM1. Image size is 100 µm × 100 µm. The TIRFM images were taken upon injection of 100 ng/mL fluorescently labeled vesicles and rCTB binding was done from a final concentration of 34.5 nM rCTB solution in a PDMS well on a cover glass.
7
S4. Theoretical background A. Maximum number of bonds The radius of SV40 is R ' 20 nm. It contains 360 capsid proteins (Np = 360) which are organized in an icosahedral structure by 72 pentamers. The number of binding sites at each pentamer is about 5, i.e., the total number of binding sites is equal to Np , and accordingly the area per site is σs = 4πR2 /Np . If the membrane is flat (this assumption is reasonable for SLB, because the membrane-support interaction is expected to prevent appreciable deformation of the membrane), the contact area which can be reached by GM1 during the SV40-SLB contact is given by σA = 2πlR [9] (the subscript A corresponds here and below to SV40), where l ' 1 nm is the length of the GM1 hydrophilic headgroup above the SLB. The corresponding number of binding sites is nA = σA /σs = lNp /2R ' 9. In reality, the membrane contacting SV40 can be somewhat deformed and nA may be larger than 9. According to [10], nA may reach 40. In our estimates in the main text, we use nA = 10. The increase of nA up to 40 does not appreciably influence our estimates and conclusions. The radius of rCTB is 3 nm, and according to the estimates similar to those given above for SV40 it can bind up to 5 GM1, i.e., nB ≤ 5 (the subscript B corresponds here and below to rCTB). B. General kinetic equations Let us consider that the nanoparticles of two types, A and B (e.g., SV40 and rCTB), attach to and/or detach from a supported membrane via receptors (e.g., GM1) located there, and each particle can form up to nA and nB bonds, respectively. In the multivalent situations we are interested in, nA and nB are typically large while each bond is relatively weak. The kinetics of the formation and rupture of such bonds are in general described in terms of surface concentrations of particles, cA,i (1 ≤ i ≤ nA ) and cB,i (1 ≤ i ≤ nB ), forming all the possible numbers of bonds, their bulk concentrations, CA and CB , and surface concentration of unbound receptors, cr . The total surface concentration of particles are given by cA =
nA X i=1
cA,i and cB =
nB X
cB,i .
(2)
i=1
The total surface coverage is defined as θ = σA cA + σB cB + σr cr ,
(3)
where σA , σB and σr are the contact areas per particle or receptor. Bearing in mind our experiments, we consider that θ 1 and accordingly treat the adsorbed overlayer as an ideal 2D solution. In this case, the equations for cA,i with i = 1, 1 < i < n1 and i = n1 can, respectively, be written as (cf., e.g., Refs. [11, 12]) dcA,1 /dt = ka cr CA (0) − kd1 cA,1 − κa (nA − 1)cr cA,1 + 2κr cA,2 , 8
(4)
dcA,i /dt = (nA − i + 1)κa cr cA,i−1 + (i + 1)κr cA,i+1 −(nA − i)κa cr cA,i − iκr cA,i ,
(5)
dcA,nA /dt = κa cr cA,nA −1 − nA κr cA,n1 ,
(6)
where ka is the rate constant of attachment of unbound A particles to receptors, C1 (0) is the concentration of these particles near the membrane, kd1 is the rate constant of the rupture of the last bond accompanied by A detachment, κa and κr are the rate constants of the bond formation and rupture occurring via the receptor attachment to and detachment from binding sites (the rates of these processes are considered to be proportional to the number of vacant and occupied binding site, respectively). The equations for cB,i are similar to (4)-(6), while the balance of receptors is described as dcr /dt = Wr,A + Wr,B ,
(7)
where Wr,A = kd1 cA,1 − ka cr CA (0) +
nA X
iκr cA,i −
i=1
nX A −1
(nA − i)κa cr cA,i
(8)
i=1
is the term responsible for binding and unbinding with A, and Wr,B is a similar term related to B. The bulk A and B concentration satisfy Fick’s second law complemented by the terms taking the solution flux into account. Near the membrane, the boundary conditions for these concentrations are obtained by considering the A and B diffusion flux in the normal direction to be equal to ka CA (0)cr −kd1 cA,1 and ka CB (0)cr −kd1 cB,1 , respectively. In general, the equations introduced above can be solved only numerically. The complicating factor is here that the equations contain many parameters which are often unknown or cannot be accurately determined. In addition, the integration of Fick’s equations under e.g. flow condition in the QCM channel is far from straightforward. Despite these complications, the equations above form a basis allowing one to interpret the kinetics under consideration. In fact, an analysis of specific systems can often be focused on the regimes when the equations can be appreciably simplified. In our context, it is instructive to discuss two such regimes. C. Initial stage of adsorption At the initial stage of the process, the concentration of unbound receptors may be appreciable, and after the formation of the first bond by A or B the formation of other bonds may be fast. Under such circumstances, the detachment of A or B will be slow, and accordingly the A or B attachment will dominate. Depending on the receptor concentration, this process may or may not be globally controlled by 9
A or B diffusion in the solution. To be specific, let us consider A attachment. If the formation of the first bond is sufficient for attachment, the rate of kinetically limited attachment is given by W a = k a cr C A ,
(9)
where CA is the A concentration in the solution (in the absence of diffusion limitations, this concentration is nearly the same far and near the membrane). The rate of diffusion-limited attachment can be represented as Wa = ka∗ CA
(10)
where CA is the A concentration in the solution far from the membrane, and ka∗ is the corresponding rate constant. The diffusion-controlled regime is realized if ka cr is comparable to or larger than ka∗ . If ka cr ka∗ , the process will occur in the kinetically controlled regime. If the concentration of unbound receptors is not sufficiently large in order to form additional bonds just after formation of the first bond, the kinetically limited attachment can be limited by the formation of e.g the first two or three bonds. In such situations, the apparent attachment rate will be nonlinear with respect to cr . If, for example, the attachment occurs primarily via the formation of the first two bonds, its rate given by solving Eqs. (2)-(7) can be represented as Wa =
κa ka c2r CA . kd1 + κa cr
(11)
To calculate the rate of diffusion-limited attachment, Fick’s equation should in general be integrated numerically. Two generic situations allowing analytical treatment are realized when the flow is negligible and appreciable, respectively. The equations corresponding to the ”no-flow” case can be found in textbooks (e.g., [13]) and original articles (e.g., [14]). Here, we are interested in the situation when the flow is appreciable. In this case, the kinetics is analytically tractable provided that the flow cell (of length L) is long and the channel cross section is rectangular (with sizes h L and l L) or circular. If, for example, the channel cross section is rectangular with h l and the A adsorption takes place on one (or both) of the l × L walls, its rate is given by [15, 16] v◦ D2 Wa = 0.97 hx
!1/3
CA ,
(12)
where x (0 < x < L) is the coordinate along the channel, D is the diffusion coefficient, and v◦ is the average flow velocity. In the QCM case, this expression can be used for rough estimates, because the flow conditions (geometry) are there more complex. 10
D. Desorption kinetics At the late stage of the process, the receptors are primarily bound by A and/or B, the concentration of unbound receptors is low, and accordingly the A and/or B adsorption may be negligible. Under such conditions, one can observe A or B detachment. To simplify Eqs. (2)-(7) in this case, we take into account that each receptor-particle bond is relatively weak and their redistribution among adsorbed A and B particles is rapid. In this case, the populations cA,i and cB,i are close to equilibrium, and instead of these populations we can operate with the total surface concentrations, cA and cB , and the probabilities, PA and PA , that a binding site which geometrically can contact a receptor is in contact with a receptor. To relate PA , PB and cr , we view each binding site, which geometrically can contact a receptor, and the adjacent membrane area where this receptor can be located as an elementary subsystem and use the grand canonical distribution in order to describe its states. This subsystem can be in three states: (i) the adjacent area is vacant, (ii) the adjacent area contains a receptor which forms a bond with the binding site, and (iii) the adjacent area contains a receptor which does not form a bond with the binding site. We are interested in the situations when the binding energies for single receptor-site pairs, EA and EB (these energies are set to be positive), are small but not too small or, more specifically, EA > kB T and EB > kB T . In such situations, the probability to be in state (iii) is low compared to the probability to be in state (ii), and accordingly state (iii) can be excluded from the analysis, i.e., in fact we have a two-state system. In this case, the grand canonical distribution yields PA =
exp[(µ + EB )/kB T ] exp[(µ + EA )/kB T ] and PB = , 1 + exp[(µ + EA )/kB T ] 1 + exp[(µ + EB )/kB T ]
(13)
where µ = kB T ln(acr )
(14)
is the chemical potentials of unbound receptors (a is the area comparable with that of the receptor cross section; in our case, a ' 1 nm2 and acr 1). Eqs. (14) and (13) can be rewritten as PA =
cr cr and PB = , K A + cr K B + cr
(15)
where KA = κr /κa = a−1 exp(−EA /kB T ) and KB = a−1 exp(−EB /kB T ) are the equilibrium constants. The balance condition for the number of receptors is cr + nA PA cA + nB PB cB = c◦r ,
(16)
where c◦r is the total concentration of receptors or, in other words, the concentration of receptors in the absence of particles. 11
To calculate the detachment rate e.g. for A particles, we take into account that the probability that an attached particle has i bonds can be expressed via the binomial coefficients as Pi =
nA ! PAi (1 − PA )nA −i . (nA − i)!i!
(17)
The derivation of this conventional expression is based on summation of all the states including that with no bonds. Strictly speaking, the latter state should be excluded from the summation because the particles reaching this state are able to leave the surface. In the multivalent case (when nA is large and PA is not too low), the probability of this state is negligibly low, and expression (17) is acceptable. According to this expression, the probability that a particle has only one bond is given by P1 = nA PA (1 − PA )nA −1 .
(18)
The surface concentration of such particles is P1 cA . The detachment rate can be obtained by multiplying this concentration by the rate constant of the rupture of this last bond, kd1 , i.e., Wd = kd1 nA PA (1 − PA )nA −1 cA .
(19)
In applications of this expression, nA can be included into kd1 , and it can be rewritten as Wd = kd1 PA (1 − PA )nA −1 cA .
(20)
This expression shows that the effective rate constant for A detachment can be defined as kd = kd1 PA (1 − PA )nA −1 .
(21)
With increasing cA and/or cB , cr decreases [Eq. (16)], PA also decreases, PA (1 − PA )nA −1 typically increases (if PA is not too low as it is implied in the analysis), and accordingly the detachment rate or rate constant increases. In applications of Eq. (20), we need to estimate kd1 . An accurate estimate of this rate constant is hardly possible. In the case of weak bonds and nm-sized nanoparticles, the rough scaling estimate of kd1 , based on the conventional Kramers theory [17], is D/a, where D ' 10−6 cm2 /s is the scale of the viscosity-related diffusion coefficient, and a ' 1 nm2 is the already introduced scale of the receptor cross section. Using these values yields kd1 ' 108 s−1 .
12
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