WO2008080417A1

WO2008080417A1 - A method of determining characteristic properties of a sample containing particles

Info

Publication number: WO2008080417A1
Application number: PCT/EP2006/012574
Authority: WO
Inventors: Mathis Harald; Jörg ACKERMANN
Original assignee: Flult Biosystems Gmbh
Priority date: 2006-12-28
Filing date: 2006-12-28
Publication date: 2008-07-10
Also published as: CN101627298A; WO2008080612A1; JP5023157B2; JP2010515031A

Abstract

A Method of determining characteristic properties of a sample containing particles of single species which emit, scatter and/or reflect photons in an predetermined observation volume including the steps of: 1) registering and counting the number ni (counting rate) of photon events registered in subsequent time inter-vals ΔtI = [ti=1,ti) (i = 1,2,3,&mldr;) of an observation time (Formula (I)); 2) determining a distribution function p(n) of the number n of photon events in a predetermined time interval Δt; and 3) using theoretical relations between the expected distribution function p(n), the concentration c, a single particle distribution function P1(n) of the number n of photon events in a predetermined time interval Δt expected in a Gedankenexperiment in which each photon event originates solely from a single particle, and an effective volume Veff =(m)/c, where (m) is the average number of particles contributing to each counting rate, for determining P1(n), Veff, the concentration c and/or other characteristic properties which can be determined from Veff and/or P1(n) by fitting these properties to the measured p(n).

Description

A method of determining characteristic properties of a sample containing particles

The invention refers to a method of determining characteristic properties of a sample containing particles of a single species which emit, scatter and/or reflect photons in an observation volume. Furthermore, the invention also extends to determining characteristic properties of a sample containing a mixture of several different species of particles.

The above methods are based on fluorescence fluctuation spectroscopy (FFS) [c.f. L. Mandel: Fluctuations of photon beams and their correlations, Proc. Phys . Soc. 72, pages

1037-1048, (1958)] which has been established as a standard method in a broad field of applications especially in biophysics and biochemistry. A theory and realization developed to be applied in high throughput screening and on-line analysis is the fluorescence-intensity distribution analysis (FIDA) known from WO 98/16814 and Peet Kask, Kaupo Palo, Dirk Ullmann, and Karsten Gall: Fluorescence-intensity distribution analysis and its application in biomolecular detection technology, Proc. Natl. Acad. Sci. USA 96, pages 13765-13761, (1999) .

In single molecule experiments the situation is as follows: Photons emitted from molecules in a small observation volume V hit a detector for photons. Discrete time points /,,/ = 0,1,2... divide the observation time into equidistant intervals

At₁ =[/,__],/,) (i = 1,2,3...N) . The numbers of photons counted in the time intervals At₁ present a sequence of numbers {n_],n₂,n_i,...} . A completed measurement gives a finite number of counting rates n_λ,n₂,...,n_N . The number of time intervals (#k) in which n photons are counted gives the photon counting distribution

Dividing each element of the sequence #k(n) by the total

number of intervals gives a real number

#k{ή)

/>*(") = N

which can be interpreted - at least in the limit of a "great many" time intervals measured - as the probability to measure n photon in a time interval At . The corresponding physical probability distribution is defined by

p(n) = lim p_N («)

Λ'→∞

For simplicity we assume all photons to be produced by molecules of interest and ignore any background noise produced e.g. by the detector hardware or scattered light. Incorporating background noise is straight forward and its influ- ence on p{n) will be discussed later.

The method of determining characteristic properties of samples in FFS includes two basic steps:

1) Registering and counting the number H₁ (counting rate) of photon events registered in subsequent time intervals At₁ =

of an observation time

N

T = ∑Δt, ι=l

2) determining a distribution function p{ή) of the number n of photon events in a predetermined time interval At

3) determining one or several properties of interest in a way that a given theoretical model describes the out- come of the measured p(n) with optimal agreement ( in the following called "fitting") .

Typical properties which can be extracted from p(ή) include

1. the concentration of molecules in pure solutions

2. the concentration of molecules in mixtures

3. the kinetic of intermolecular binding processes

4. the kinetic rates of intermolecular dissociation processes 5. the kinetic rates of intramolecular conformal changes

6. the diffusion rates of molecules

7. the molecular brightness

8. the spatial brightness function of the optical set up

9. the life time of singlet states specific to molecules

10. the after-pulse rate of a detector

In FIDA the theoretical photon count distributions p{ή) is

computed via the generating function G(ξ) = y]p(n)ξ". G(ξ) is n=0 expressed as the exponential of spatial integrals over a function depending on a so-called spatial brightness function B(r) . The spatial brightness function is the product of the excitation light intensity and transmission coefficient of fluorescent light by the optical equipment as a normalized function of coordinates of a particle in the sample. A simple model for B(r) is applied to characterize the optical equipment, and adjustment parameters of B{r) are determined by experiments on single species. The unknown model parameters, the concentration c and a specific brightness value q, are determined either by a non-linear fitting procedure or by an inverse transformation with regularization. Disadvantages of FIDA are

1. The generating function approach makes the theoretical model not very intuitive, and the extension of the method to more complex applications is difficult.

2. The model is highly non-linear in the unknown parameters c and g and the determination of these parameters is complicated.

3. The brightness of each molecule is expressed as a product of a spatial brightness function, which is common to all species, and a specific brightness, which has a characteristic value for each species. Since triplet state population differs significantly for different species this assumption is violated to some extent in all measurements.

4. The coordinates of the molecules are assumed not to change during the counting time interval (bin width) .

Even for a very short time interval in the range of a few μs this is only a crude approximation. Moreover the dependence of p{n) on the diffusion movements of the molecules bears useful information which cannot be ex- tracted from the model.

By the introduction of semi-empirical correction factors in FIDA a so-called fluorescence intensity multiple distribution analysis (FIMDA) has been developed for a concurrent determination of diffusion times and molecular brightness [c.f. Kaopo Palo, UIo Mets, Stefan Jager, Peet Kask, and

Karsten Gall: Fluorescence intensity multiple distribution analysis: Concurrent determination of diffusion times and molecular brightness, Biophysical J. 79, pages 2858-2866, (2000)]. Compared to FIDA, the direct calculations of photon count distributions via Master equations show a significantly improved fit quality. Such numerical simulations, however, are very slow and not suited for high throughput applications [c.f. Kaopo Palo, ϋlo Mets, Velio Loorits, and Peet Kask: Calculation of photon count number distributions via master equatons, Biophysical J. 90, pages 2179-2191, (2006) ] .

Methods alternative to FIDA and its successors are based on the photon count histogram PCH algorithm [c.f. Yan Cheng, D. Mϋller Joachi, Peter T. C. So, and Enrico Gratton: The photon counting histogram in fluorescence fluctuation spectroscopic, Biophysical J. 11, pages 553-557, (1999), Yan Cheng: Analysis and applications of fluorescence fluctuation spec- troscopy. PhD thesis, University of Illinois at Urbana- Champain, Urbana, Illinois, (1999), available at http://www.lfd.uiuc.edu/staff/gratton, Thomas D. Perroud, Bo Huang, and Richard N. Zare: Effect on bin time on the photon counting histogram for one-photon excitation, ChemPhysChem 6, pages 905, 912, (2005), Y. Cheng, M. Tekmen, L. HiI- lesheim, J. Skinner, B. Wu, and J. Mϋller: Dual-color photon counting histogram, Biophysical J. 88, pages 2177-2192, (2005)]. These methods differ only in technical details from FIDA and assume a different shape for the spatial laser brightness distribution. PCH methods play a minor role for commercial applications. In the PCH approach the detection volume has to be chosen rather arbitrarily. Properties like the average number of contributing molecules and the photon probability distribution of a single molecule become ab- stract properties without any physical meaning. The object of the present invention is to provide an improved theoretical platform for the a priori prediction of photon counting histograms. A further object of the inven- tion is to provide a more profound physical insight into FFS and to make the PCH approach intuitively extendable to complex experimental tasks.

The above objects of the present invention are solved by the method having the features of claim 1.

The effective volume V_eff is not readily described by the optical set up but also depends on the properties of the particle species considered. A particle is defined to be "inside" this volume during a particular time interval if and only if it contributes to the count rate. The probability to contribute to the count rate is non-zero for a particle located at any space position. Hence the effective volume V_eff cannot be determined by physical spatial boundaries. Nevertheless the process of "entering" and "leaving" this volume is a stochastic process similar to the diffusion process into and out of a small physical volume. Note that techniques like stimulated emission depletion (STED) [c.f. Lars Kastrup, Hans Blom, Christian Eggeling, and Stefan W. Hell: Fluorescence fluctuation spectroscopy in subdiffraction focal volume, Phys . Rev. Lett. 94, (2005)] are capable to reduce significantly the effective volume.

By definition, a "single particle" inside the effective vol- ume cannot produce any count rate of zero. Consequently the term to have a "single particle" inside the effective volume is conceptionally very different from having a "single particle" in any spatial volume. Physically the probability distribution P_\{n) can be measured in a Gedankenexperiment by considering only non-zero count rates in which all photon events during a time interval Δt, originate solely from a single particle. Superposition of signals originated from several particles has to be ignored.

According to the present invention, a particle species, e.g. a molecular species is characterized for a given experimental set up by two quantities: the effective volume V_eff and the single particle distribution P,(«) . Both quantities have a defined physical meaning, as discussed above. The single particle distribution P_x{n) can be distinguished from the identically named single particle distribution in the FIDA and PCH approaches by its unique property P₁(^ = O) = O.

The knowledge of V_eff and P_\{ri) can be utilized for a fast and robust determination of molecular concentrations in samples containing one or several species (mixtures) of molecules. The method is robust against disturbing effects as e.g. after-pulsing, molecular diffusion or singulett state excitation which limits the applicability of all previous methods as e.g. FIDA, FIMDA, or PCH algorithms. The method according to the invention is not limited as FIDA or PCH algorithm to short bin widths. It is applicable to any molecular environment as e.g. flows, micro-structures, cells, vesicles, emulsions, or gels (not experimentally verified up to now) .

The method according to the present invention is more simple than all previous methods. Technically it can be combined with standard fitting methods as e.g. non-linear fits, generating functions, or method of moments. Depending on the technical realization it is a very fast method suitable for on-line diagnostics.

In FIDA a kind of molecule is characterized by a specific molecular brightness given by a real number value. According to the invention, a kind of molecule is fully characterized by V_ejr and i» .

The properties V_eg and P₁(^) are rich in the sense that they contain all information as e.g. diffusion rates or singlet state excitation probabilities relevant in the experimental set up. In principle all information characterizing the particles, e.g. molecules, can be extracted from V_eff and P_x{n) .

To get this information, however, a specific theoretical model has to be applied.

The properties V_eff and P_λ(ή) are pure in the sense that they characterize the properties of a single particle without any averaging process present in any experiment due to the si- multaneous contributions of several particles.

In a preferred embodiment of the present invention, the single particle distribution function P_x(n) , based on the theory of Markov processes, is given by

with /^>(« = 0):=0,

wherein μ(f) denotes the brightness function of the particle defined by the average value of photon events by a single particle at position F , and the Poissonian distribution Poi(n,μ) = exp(-//)//" In\ .

A photon count rate Yi₁ for a given bin width At₁ depends strongly on the positions of all molecules relative to the laser focus because of the spatial varying laser intensity and the photon collection properties of the optical set up. Molecules near the laser focus produce a high contribution to the photon count rate, whereas molecules far away from the laser focus give rise to a low or no contribution. The contribution of a single molecule can be measured by immobi- lizing it at a given position r, on a surface (e.g. glass surface) or in a matrix (e.g. gel). The series of photons counts H₁ recorded for this set up gives the probability distribution p{ri) . The so measured average value of photon counts (n) = ^_jnp(n) is called molecular brightness μ at posi- n tion (F) . The measurement of the molecular brightness μ{r) is possible for any position F , and the functional form of μ(f) can be measured by repeating this procedure for various locations of the molecule, e.q. by measuring μ for all positions F on a equidistant spatial mesh and interpolating μ{r) between the mesh points. Since the measurement of the molecular brightness function μ{r) is time consuming and expensive, it is usually approximated by models based on the theory of molecular fluorescence spectroscopy and given models of the spatial laser light intensity. In this context, the molecular brightness μ(r) depends on parameters as e.g. the bin width At , the spatial distribution and intensity of the light at the excitation wavelength (λ) , the collection efficiency function of the optical set up, the quantum efficiency of the detector (q) , the cross section (σ) , and the fluorescence quantum yield (φ_f) of the given type of molecule. Effects like triplet state excitation and the after- pulsing of the detector have been shown to play an important role as well.

The knowledge of the molecular brightness μ(r) is inevitable for any ab initio prediction of experimental photon count distributions. In FIDA and PCH, theoretical models for //(F) are applied. Free parameters in these models are determined by experiments on single species.

If it is assumed that the molecular brightness function μ{r) is known either by direct measurements or an approximation by a theoretical model, the probability P_x{n) to measure n photons from an individual molecule is described by Mandel^'s classical formula [c.f. L. Mandel: Fluctuations of photon beams and their correlations, Proc. Phys . Soc . 72, pages 1037-1048, (1958) ] :

P,{n)=[Poi{n,μ{7))p{7)dV

where the Poissonian distribution Poi{n,μ) = exp(-μ)μ" In\ is averaged over all possible locations of the molecule. p{r)-\IV denotes the constant probability to find the particle at the space point F . Unfortunately the above equation for P_x(n) = J Poi{n,μ(r))p(r)dV results in the simple solution

P_\{n) = δ_On in the limit of infinitely large integration volumes V. This reflects the physical fact that an individual molecule in an infinite volume corresponds to a zero concentration and cannot give rise to any photon count event at all. A common practice is to circumvent this drawback by introducing a large but finite volume V, see, for example, the discussion in Yan Cheng, Joachim D. Muller, Peter T. C. So, and Enrico Gratton: The photon counting histogram in fluorescence fluctuation spectroscopie, Biophysical J. 11, pages 553-557, (1999) .

According to the present invention, the above-mentioned problem of the PCH algorithm is avoided by applying the definition of the "single molecule" in the effective volume V_eff to Mandel's formula which then reads: P_λ(ή) = Ϊ/V_ejr J, Poi(n, μ(r)) dV ,

with P₁ (H = O) I= O .

This equation establishes a relation between P_λ(ή) , V^ and

//(F) which may be used for determining one or the other of these characteristic properties.

Furthermore, the Poissonian approximation, based on the theory of Markov processes, leads to the following formulae for

(m) and V_cf/ .

Taking into account non-zero concentrations c in real physi- cal situations, the average number (ni) of molecules contributing to each counting rate is given by

where cdV denotes the number of molecules in volume element dV and the factor

describes the probability of each molecule in dV to give rise to at least one photon count event .

By means of the relation V_eff:=(rnjlc one obtains

V_eff = [\\-Poi(n = Q,μ(7))-\ dV .

The above equations establish theoretical relations between (rrή, //(F) and c, as well as between V_ejr and //(F), respectively, which may be used for determining one or the other of these characteristic properties. In most applications of FFS, the particles (molecules) of the same species are fluorescent or fluorescently labelled, and a laser beam is used for the excitation of the parti- cles.

In theoretical considerations, the most popular model for the brightness function is the spatial Gaussian profile

/α²),

where μ_max is the brightness of a molecule in the center of the laser focus and a denotes the waist parameter of the laser beam. The limitation of such a bold approximation to account for the rather complex three-dimensional spatial brightness function has been demonstrated by several groups, see Bo Huang, Zhomas D. Perroud, and Richard N. Zare, Photon counting histogram: One-photon excitation, ChemPhysChem 5, pages 1323-1331, (2004) and literature therein. A few re- marks are appropriate to justify this choice as an illustrative example. Firstly, the present invention our approach may be applied to any spatial brightness function. Secondly, approximations by Gauss functions are usually easy to handle and contractions of Gauss functions have been proven to con- verge to any integrable spatial function [c.f. Bruno Klahn and Werner A. Bingel: The convergence of the Raleigh-Ritz Method in Quantum Chemistry, Theoret. Chim. Acta 44, pages 9-43, (1977)]. This is one of the reasons why they have been applied extensively in modern quantum chemistry for decades.

Inserting the Gaussian profile into the definition of the effective volume and the single molecule function, respectively, a transformation of coordinates r —» μ , and the integration over the rotational symmetry yield

and

with

and

Numerical Romberg integration may be applied to compute the functions F and Q. Note, that Q is finite only for n≥\ and diverges for « = 0. For a Gaussian shaped μ{r) the effective volume V_eff becomes identical to the corresponding detection

volume in FCS V_fcs=π²ct³ for F(μ_max ) = *j2π , i.e. for certain values of μ_mx , P_t(n) can be calculated in a numerically stable and fast way for any /Z_1113x and has a defined physical meaning.

The theoretical computation of the single molecule function P_λ{n) and V_eff as described above is based on the classical

Mandel formula. This formula, however, has certain limitations. The distribution of the photons count rates may devi-

ate from a Poisson distribution Poiin,μ{r)) , because the rate of production of photons may vary during the time interval Δ/ . Several effects may cause such a variation of the photon production rate. A single molecule, starting at time point / at position r , may move and the molecule may visit other locations during the time interval At . Since the spatial laser intensity varies, this diffusion process, for example, may lead to a significant non-constant photon production rate and a break down of the Mandel formula. In such cases the Poisson distribution in the formula of Mandel has to be replaced by a more general distribution p(n,r)

for n > 0 and P, (« = 0) := 0 . The ef fective volume is given by

K_jr ^■■ = I (\ - p(n = 0,r)) dV . p(n,r) is the distribution of the number of photon events counted during the time interval te [t_o,t_o+Δt) for a single molecule located at time point t₀ at position F . The computation of the distribution p(n,r) , in general, is a theoretical task. To account for the diffusive motion of the molecule, the distribution p(n,r) may formally be written in the form:

N

/^>oz-(«,l /(W + l)∑ t )) ι Σ>f =0

Where G denotes the Greens function of the diffusion

In the limit of slow diffusion D -> 0 the traditional Mandel formula is obtained via lim P(ⁿ _>ϊ) = Poi{n,μ{7))

D→O

This example shows how the concept of a single particle dis- tribution function P_λ(n) introduced above can easily be extended to complex measurement conditions. It should be noted, however, that P^n) and V_eff may be determined experimentally in cases where theoretical considerations are not reachable .

Starting from the single particle probability distribution

P_x(n) , the connection between P₁(H) and a measured distribution function p(ή) is obtained as follows.

In real physical situations, the average number of (rnj = cV_eff particles contributes simultaneously to the signal. Making use of the theory of Markov processes, the probability distribution p{n,c) for a given concentration c is determined by the summation over all m-particle contributions

p{n,c) = ∑Poi(m,{m))P_m(n) ,

ΛI=0

where P_m{n) denotes the mth convolution of the single particle probability distribution P_x{n)

n-1

P_m(n) = YP_m_,{n-i)P_λ{ϊ) m = 2,3...,

and P_ύ(ri) = δ_nϋ. Alternatively, p(n,c) may be computed via the recurrence formula: cV " p(n,c) = -^∑iP_] (i)p(n - i,c) ,

with

p(n = O,c) = exτp(-cV_eff )

Ignoring back ground noise, the concentration of particles is completely determined by the probability to get the count rate zero:

c = -\n(p(n = 0,c))/V_ejr.

The distribution p(n,c) may have to be convoluted with a Poisson distribution

to account for additional background signals (e.g. random noise of the hardware):

_P>oXn,c,{n)_nnJ =(p{c)®Poi({n)_noJ){n).

According to the above theoretical considerations, a sample containing fluorescent particles (e.g. molecules) is characterized for a given experimental set up by the concentration c, the brightness function μ{f), the effective volume V_eff and the single particle distribution P_λ(ri) . Since a noise contribution {n)_noιse cannot be neglected in most experimental set ups it has to be considered as well. Following the convention described in FIDA and the PCH algorithms these parameters can be determined by a nonlinear multi-parameter fit procedure, e.g. a Marquard-Levenberg algorithm or any other numerical standard fit method appropriate for this task. Therefore the brightness function μ{r) has to be approximated by an analytical function with a number of adjustable parameters. The analytical form of μ(r) can be chosen as a Gaussian function, like in the above example, a contraction of several Gaussian functions or any other appropriate shape function.

All adjustable parameters of //(F), as well as the concentration c, and the noise contribution

have to be deter- mined by a non-linear fit of the theoretical model to the measured distribution p(ri) . The effective volume V_cff , and the single particle distribution P_x{n) follow directly by inte- grating μ{r) as described above.

An alternative procedure for the non-linear fit procedure is the method of moments which utilizes a pre-calculated list of moments to be expected for any set of values for the ad- justable parameters of μ{7) , the concentration c, and the noise contribution

. Since such a list can be calculated and stored in advance for appropriated ranges of the parameters, this is the fastest possible procedure for on-line applications. The moments of a distribution p(ri) are defined by

v, =∑P(Φ' ■

Let U₁ , i = \,2,...,k denote the k parameters to be determined. Within an appropriate range of values of the parameters α, the expected moments V₁ i = 1,2,...,/ can be calculated. Precalculation of the first 1 moments v for all possible sets of parameter values a_t , i = l,2,...,k yields a mapping

(a_],a₂,...,a_k)→(v_ι,v₂,...,v_l) .

A grid of discrete parameter values reduce the computational effort, and expected moments for parameter values not on this grid have to be extrapolated, e.g. by a spline extrapolation. The number of moments 1 is chosen as low as possible but sufficiently high to guarantee the above mapping to be a bijective function. The mapping can then be inverted to

(v_ι,v₂,...,v_l)→(a_ι,a₂,...,a_k)

which gives the demanded parameters for any set of measured moments V₁, /^' = 1,2...,/.

The noise contribution

can be measured directly for an experimental set up by using the medium liquid of the sample without fluorescent particles. The medium liquid of the sam- pie may be replaced by a liquid having identical or similar

(optical) properties. Determining (n) in advance, this parameter can be kept fixed in the non-linear fit procedure and the method of moments. In this way, the number of demanded parameters in these procedures is reduced by one.

The above considerations concern a sample of particles of a single species.

The present invention straightforwardly also applies to sam- pies containing N different species of particles (a mixture of particle species) .

In the case of a mixture of particle species the determination of the concentration of each species is done in a multi-step procedure. First the effective volume V^ and a single particle distribution P, («) have to be measured for each species s present in the mixture and the given experimental set up. This is done by one of the procedures described above. Based on determined properties

the expected total distribution p(n;c_],c₂,...,c_N) can be expressed by the convolution of the single species contributions

, s = 1,2,...,JV :

Parameters to be determined are the concentrations c^s.,' s = 1,'2,'...,'JV and the noise contribution

The demanded parameters can be determined by using a nonlinear fit of the measured data to the theoretical models. Similarly to the non-linear fit procedure described above, standard numerical techniques can be applied. A generating function approach may speed up the computation of p{n;c_x,c₂,...,c_N) in some cases.

Alternatively to the non-linear fitting, the parameters can be determined by the method of moments as described above. This procedure is technically simplified by the fact that the moments of p(n;c_x,c₂,...,c_N) can be expressed as products of the moments of p^(<l)(n;c_s) for s = \,2,...,N.

A measurement of V_eff and P₁(^) is possible without any model or knowledge on the molecular brightness function. These properties may be determined by an analysis of the measured distribution p(n,c) directly. Since a measured distribution is noisy data, the effect of error propagation can destabilize numerical methods for the extraction of /*,(«) . For this reason a global representation of P_λ{ri) is advantageous. A powerful method is a so-called discrete Galerkin approximation of the form i» = f>,ψ(«,jM) l_k(n,p,λ)

*=o

where a_k are generalized moments, Ψ(n,p,λ) is a weighting function with adjustable parameters p and λ, and l_k(n,p,λ) are the corresponding polynomials. The parameters p and λ are determined by the first moments of P,(«) . Such error controlled Galerkin projections are well-studied methods in numerical mathematics and usually applied to polymer chemis- try [s.f P. Deuflhard and J. Ackermann: Adaptive Discrete

Galerkin Methods for Macromolecular Processes, in H. P. Dik- shit and Charles A. Michelli, editors: Advances in Computational Mathematics, World Scientific Publishing Co., Inc., (1993); J. Ackermann and M. Wulkow: MACRON - A Program Pack- age for Macromolecular Reaction Kinectics, Konrad-Zuse-

Zentrum, Preprint SC-90-14, (1990), M. Wulkow and J. Ackermann: Numerical Simulation of Macromolecular Kinetics - Recent Developments, IU-PAC Working Party, Macro group, (1990); M. Wulkow and J. Ackermann: The Treatmeant of Macro- molecular Processes with Chain-Length-Dependent Reaction

Coefficients - An Example from Soot Formation, Konrad-Zuse- Zentrum Berlin, Preprint-91-18, (1991); U. Budde and M. Wulkow: Computation of molecular weight distributions for free radical polymerization systems, Chem. Ing. Sci. 46, pages 497-508, (1991), M. Wulkow: Numerical Treatment of

Countable Systems of Ordinary Differential Equations, Thesis and Technical Report-90-8, Konrad-Zuse-Zentrum Berlin, (1990); M. Wulkow: Adaptive Treatment of Polyreactions in Weighted Sequence Spaces, IMPACT Comput. Sci. Engrg. 4, pages 152-193, (1992)]. The parameters of this approximation can be obtained from p(n,c) either by a fitting procedure (see above-described non-linear multi-parameter fit procedure) or by a momentum method analogously to the above- described method of moments.

A useful property in this context is the averaged molecular brightness producing n photons defined by

{μ)_n characterizes the spatial brightness μ(r) in terms of effective photon production rates and is connected to the probability distribution P_\(n) through

Typically, the molecular brightness increases linearly (μ}_n&n for low count rates n and becomes constant at the maximum molecular brightness of one molecule in the center of the excitation light focus. This property makes the single molecule signal well distinguishable from random noise signal which would give a constant distribution^//^ = con- stant.

The foregoing and other objects, aspect and advantages of the invention will be better understood on the basis of the following three examples where a polystyrene micro spheres suspension series, a dye Rhodamine 6G dilution series as well as a mixture of a polystyrene micro spheres and Rhodamine 6G dye are analyzed. A prior art fluorescent spectroscopy unit comprising a light source emitting 532 nm excitation light at an intensity of 50 μW, a built-in high sensi- tive photomultiplier tube as well as a digital correlator has been employed to collect the experimental data. The fluorescent spectroscopy unit used has a reaction time of 30 nsec. Data acquisition has been accomplished by counting clock pulses between two successive photon registrations. On the basis of such collected data, several light intensity traces may be calculated for different bin widths ΔT.

Example 1 : Micro spheres suspension series

As a first example, an analysis of a fluorescent polystyrene micro spheres suspension series has been conducted.

Figure 1 shows a typical light intensity trace recorded over 3,2 sec for a sample containing polystyrene micro spheres (beads) with a molar concentration c^ 1,9E-IO Mol/L.

Figure 2 shows an autocorrelation curve (averaged over 100 traces a 3.2 sec as shown in Fig. 1) .

Figure 3 shows the corresponding photon count distribution p(n,c) for a bin width ΔT = 0,1 msec.

Figure 4 shows the numbers of molecules obtained by fluorescence correlation spectroscopy (FCS) (broken lines) as well as obtained by methods according to present invention (solid line) for various samples. As reference, the theoretical drop down slope is also shown (unconnected squares) .

Figure 1 shows a typical light intensity trace recorded for a sample containing 0,014 μm beads of known molar concentration c_jyi^% 1,9E-IO Mol/L. The bin time has been adjusted to 1 msec. The intensity peaks indicate the transition of molecules through the confocal FCS volume.

Figure 2 shows the corresponding autocorrelation curve (averaged over 100 loops a 3.2 sec) . A fit lead to a diffusion time of τ=l,74 msec and n=0,1529 beads in the confocal FCS volume. The diffusion time averaged over several FCS measurements gives τ=l,91 ± 0,33 msec. This diffusion time determines the period of time a bead requires passing through the confocal FCS volume. According to manufacturer information, the fluorescent spectroscopy unit employed has a FCS volume of V_FCS * 1 femto liter (fL) .

Figure 3 shows the corresponding photon count distribution p(n,c) for the bin width ΔT = 0,1 msec . The probability p(n=0,c)= 0,19976 means that around 80% of all photon counts in figure 1 are zero. A measurement with pure water gives the background noise level of <n>_noise ⁼ 0, 0230642.

According to the present invention, the average number of molecules <m> in the effective volume V_eff is given by above formula

c V_eff = <m> = - ln (p (n=0 , c) ) - <n>_noise = 0 , 20038.

Consequently, the effective volume V_eff according to the present invention turns out to be in the order of magnitude of (c_M = 1,9E-IO Mol/L, N_A = 6.022E23 )

V_eff = <m> / (c_M N_A ) « 1,75 fL

and hence is slightly larger than the confocal FCS volume. Note, that the value of V_Θff may be decreased (enlarged) by reducing (increasing) either the bin width ΔT or the excitation light intensity, data not shown. In general, V_βff depends on apparatus parameters as well as on molecule parameters .

In figure 4 the numbers of molecules obtained by standard FCS analysis are shown (broken lines) for various samples. Two measurements have been performed for each sample; each measurement is indicated by dots (measurement I) and stars

(measurement II), respectively. The series of measurements starts (step 1) at the initial solution of predetermined c_M =

2,44E-8 Mol/L.

This initial solution is diluted in twelve steps by a "mix and split" method. A sample of 50 μL is diluted with 50 μL water; half of the resulting 100 μL is taken for measurements and the other half serves as sample for the next "mix and split" step. Ideally, the concentration of micro spheres drop by a factor of two by each dilution step. The lowest concentration of micro spheres is reached in step> 13, resulting to c_M = 3E-12 Mol/L. In the logarithmic scale of figure 4 the number of beads obtained by the standard FCS should drop down linearly as indicated by the unconnected squares .

Small deviations from this theoretical behaviour may result from uncertainties in the handling of small volumes during the series of "mix and split" steps. However, it is obvious that the results of two measurements of the same sample conducted by standard FCS vary between 10% and 50%. This variation indicates a rather large statistical error of determined number of beads as well as a similar uncertainty of the concentration obtained by the FCS. More drastically, however, is the behaviour of FCS for low concentration. For concentrations lower than c_M =3,8E-10M the FCS method is unable to determine the correct concentration and gives erroneous high values. The FCS gives a value of two orders of magnitude too high for the sample in dilution step 13. This behaviour of FCS discussed above is well known and presents a serious limitation of the FCS method especially for low concentrations.

In the following, it will be seen, that the inventive method results in much more precise determination of the number of fluorescent particles with respect to prior art FCS method. According to the present invention the concentration of fluorescent particles, i.e. the number of particles <m> in Vg_ff, can be determined by p(n=0,c), see above. Note, that p(n=0,c) can be measured with a high statistical precision in the low concentration regime. The corresponding numbers of molecules are shown in figure 4 (solid lines).

The inventive method has been applied to the same acquisition data records used by the FCS method. Accordingly, no additional measurements have been performed to obtain these results by the inventive method. Compared to the values derived by FCS, the statistical variation of <m> is much lower (between 1% for concentrations around c_M =5E-10 Mol/L and 10% for the lowest concentration c_M =3E-12M) . More important, the measured concentration follows the correct theoretical linear drop down slope and no detection limit is visible down to a concentration of c_M =3E-12M.

Example 2 : Dilution series of the dye Rhodamin 6G

Similar results have been obtained for a dilution series of the dye Rhodamin 6G (identical conditions as above for example 1) . Here the detection limit of the FCS is determined by the sensitivity of the apparatus and is in the order of c_M =3E-9 Mol/L. The present invention turns out to enable the measurement of the concentration with precision of 2% even below the detection limit of the FCS, data not shown. For Rhodamin 6G the value for the effective volume reduces to V_eff « 0.2 fL for given apparatus parameters of the fluorescent spectroscopy unit.

In order to increase the statistical significance of the photon count signals produced by fluorescent particles it may be advantageous to choose a rather large bin width.

Compared to standard analysis methods and their successors the present invention enables the determination of very low concentrations with superior simplicity and precision.

Example 3: Mixture of Beads and Rhodamin 6G

Measurements of a sample containing a mixture of beads and dye Rhodamin 6G (RhβG) gives an average number of particles of <m> = 2,2 in the effective volume (experimental conditions as above) . This value is similar to the FCS value of <m>_FCS = 2,35. A two parameter FCS, however, is unable to determine the concentration of each species separately. According to the present invention the concentration of beads and RhβG can be determined by various methods. One way is to utilize the methods of moments.

The first moments of the single molecule distributions for beads and Rh6G, V₁ (/><*""* >) = 1,55 and V₁ (/><**^6C>) = 1,06 , are computed from the measurements described in above example 1 and example 2 , respectively. Therefore the relation

v\ (Ptot )=< »> noise + < ^{m > v}ι(^p\)

between the first moment of the measured distribution ) ^and the moment of the single molecule distribution

V₁(P₁) is applied. Note, that the values V₁(P₁ ) and

V₁(P₁ ) characterize each species independent of its actual concentration. The first moment of the single molecule signal obtained for the mixture gives

V₁(P₁ ) = 1,30 and fit neither to the moment of the beads nor to the moments of the dye. To compute ., / _D (beads +Rh 6G) \ .

V₁C-H ) ^the relation

_/ ( beads + Rh 6 G ) N

"l (P,« ) =< « > noise +

(< m >^<Ws > + < ι» ><^JB«^σ>)* V₁(P₁"^^{1 +Λ}"^6G))

is applied. The portion x :=< m >^(iM* > /(< in ><*"* > + < m ><^Λ*^6G>)

of beads contribution to the signal in the mixture is easily determined by the relation

_{/ Tt} (beads +Rh 6G) \ , _{/ o} (beads ) >. , _/-. N. , , j-iiRhβG)

V₁(P₁ ) = x* V₁ (P₁ ) + (1 - X) * V₁(P₁

We obtain

<m>beads = x*<m> = 0,95 <m>_Rh6G = (l-x)*<m> = 1,25

The resulting concentrations (in mol/L) are

c_M (beads) = <m>_beads / (V_eff (beads) *N_A) * 1E-9 Mol/L

c_M (RhβG) = <m>_Rh6G /(V_eff (RhβG) *N_A) * 1E-8 Mol/L

The concentrations measured for a dilution series of this sample reproduce the correct concentration drop down for both species, data not shown.

A alternative method to determine the ratio x is the comparison of the first nonzero components of the single molecule distribution P₁(H=I) of beads, RhβG and the mixture, respectively. The relation

P_tot (l) = <m> exp (-<m> -<n>_noise) P₁ (I ) + <n>_noise exp (-<n>_noise)

has been applied to obtain the single molecule distribution Pi(I) from the measured p_tot(n=l). The ratio x is determined via x_{= [jPi}(i)<"⁺™^6G>_ P₁(I)^^60J]Zt P₁(Iy*"**- P,(1)^(/?A6G)]

leading to a value of x * 0.36±0.15. Both methods described above enable the distinction of mixtures from pure solutions and to determine the order of magnitude of the concentration of each species in the mixture.

Claims

1. A method of determining characteristic properties of a sample containing particles of single species which emit, scatter and/or reflect photons in an observation volume including the steps of:

1) registering and counting the number H₁ (counting rate) of photon events registered in subsequent time intervals At₁ = [/,_,,t_: ) (/ = 1,2,3...)of an observa-

N tion time T = ∑At₁ ,

2) determining a distribution function p{n) of the number n of photon events in a predetermined time interval At and

3) using theoretical relations between the distribution function p(n) , the concentration c, a single particle distribution function P_x(ri) of the number n of photon events in a predetermined time interval At expected in a Gedankenexperiment in which each photon event originates solely from a single particle, and an effective volume V_eff:=(m)/c , where (rri) is the average number of particles contributing to each counting rate, for determining P_λ{n) , V_ef, , the concentration c and/or other characteristic properties which can be determined from V_11J1 and/or P₁ (n) by fitting these properties to the measured p(n) .

2. The method according to Claim 1, wherein the single par- tide distribution function P₁(_^n), based on the theory of

Markov processes, is given by

P_x {n) = \I V_elf \_e Poi{n,μ{7)) dV,

with P₁ (W = O) = O ,

wherein μ(r) denotes the brightness function of the particles defined by the average value of photon events by a single particle at position r , and the Poissonian distribution Poi(n,μ) = exp(-μ)μ" I n\ .

3. The method according to Claim 1 or 2, wherein the brightness function μ{r) is determined by measuring μ for a plurality of positions F and interpolating μ be- tween these positions.

4. The method according to Claim 1 or 2, wherein the brightness function μif) is determined by theoretical models based on the theory of molecular fluorescence spectroscopy and given models of the spatial laser light intensity.

5. The method according to one of the Claims 1 to 4, wherein P^n) is determined by the brightness function μ{r).

6. The method according to one of the Claims 1 to 4, wherein V_eβ- is determined by the brightness function μ(r).

7. The method according to Claim 1 or 2, wherein the brightness function μ(r) is determined by P^{ή) or V_eff .

8. The method according to Claim 7, wherein P_{(n) or V_eff are measured directly.

9. The method according to one of the Claims 1 to 8, wherein the particles of the same species are fluorescent or fluorescently labelled and are excited by a laser beam.

10. The method according to Claim 9, wherein the brightness function μ(r) is modelled by a spatial Gaussian profile

where μ_mai is the brightness of a particle in the center of the laser focus and a denotes the waist parameter of the laser beam.

11. The method according to Claim 9, wherein the brightness function μ(r) is modelled by a superposition of Gaussian profiles

with parameters μ_t and a_} (j = 1,2,...,K) .

12. The method according to Claim 1, wherein the single particle distribution function P_x(ή) is given by

Wherein P{n,r) is the distribution of the number of photon events counted during a time interval /e

[t_o,t_o+Δt) for a single molecule located at time point t₀ at position r .

13. The method according to one of the Claims 1 to 12, wherein, based on the theory of Markov processes, the theoretical relation

p(n,c) = ∑Poi(m,(m))P_m(n), m=0

between the probability distribution function p(n,c) and the single particle distribution function P_\(ri) is used,

where P_m(n) denotes the mth convolution of the single particle probability distribution P_λ{ri)

n-\

P_nAn) = YP_m__x{n-Z)P₁(O, in = 2,3

(=1 and P₀(n) = δ_n0.

14. The method according to one of the Claims 1 to 13, wherein the probability distribution function p{n,c) is computed via the recurrence formula:

with

p(n = 0,c) = exp(-cV_eff)

15. The method according to one of the Claims 1 to 14, wherein the concentration c of particles is deter- mined by the probability to get the count rate zero:

c = -\n(p(n = 0,c))/V_eff .

16. The method according to one of the Claims 1 to 15, wherein the characteristic properties according to step (3) of Claim 1 are determined by a standard non-linear multi-parameter fit procedure.

17. The method according to one of the Claims 1 to 15, wherein the characteristic properties according to step (3) of Claim 1 are determined by the standard method of moments.

18. The method according to one of the Claims 1 to 17, wherein the distribution function p(n,c) is determined through convolution with a Poisson distribution

Poi(n,(rij ) to account for additional background sig- nals (e.g. random noise of the hardware):

® Poiti \\n) l _mnse)")(^/*) ' .

19. The method according to Claim 18, wherein the noise contribution («) is measured directly by using the medium liquid of the sample without any particles of said species or by using a liquid having identical or similar properties as the sample, to keep (n)_nmse fixed in step (3) of Claim 1.

20. A method of determining characteristic properties of a sample containing a mixture of N different species of particles , which emit, scatter and/or reflect photons in an observation volume, wherein the overall distribution function p{n;c^,c₂,...,c_N) of the mixture is determined by the convolution of the single species contributions p^(s)(n;c_s), s = l,2,...,N:

P(HJC₁₉C₂,...,^) = p^w{n;c_x)® p⁽²⁾{n;c₂)®---® p^(N\n;c_N)

whereby the single species contributions p (n;cj are determined by the predetermined effective volumes V^ and single particle distributions P^(n) for each species s present in the mixture and the given experimental set up, according to one of the Claims 1 to 17.

21. The method according to Claim 20, wherein the parameters c_o ^s '

are determined by using an non-linear fit of the predetermined parameters to the theoretical models.

22. The method according to Claim 20 or 21, wherein the computation of p(n;c_],c₂,...,c_N) is speeded up by a generating function approach.

23. The method according to Claim 21, wherein the demanded parameters are determined by the method of moments whereby the moments of p(n;c_λ,c₂,...,c_N) are ex- pressed as products of the moments of

for s = \,2,.,N .