US20160161390A1 - Method for identifying and quantifying of emitting particles in systems - Google Patents

Method for identifying and quantifying of emitting particles in systems Download PDF

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US20160161390A1
US20160161390A1 US14/903,784 US201414903784A US2016161390A1 US 20160161390 A1 US20160161390 A1 US 20160161390A1 US 201414903784 A US201414903784 A US 201414903784A US 2016161390 A1 US2016161390 A1 US 2016161390A1
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particles
species
function
time
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Benjamin Greiner
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Fraunhofer Gesellschaft zur Forderung der Angewandten Forschung eV
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/62—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light
    • G01N21/63—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light optically excited
    • G01N21/64—Fluorescence; Phosphorescence
    • G01N21/6408—Fluorescence; Phosphorescence with measurement of decay time, time resolved fluorescence
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume or surface-area of porous materials
    • G01N15/10—Investigating individual particles
    • G01N15/14—Optical investigation techniques, e.g. flow cytometry
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/62—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light
    • G01N21/63—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light optically excited
    • G01N21/64—Fluorescence; Phosphorescence
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume or surface-area of porous materials
    • G01N15/10—Investigating individual particles
    • G01N15/14—Optical investigation techniques, e.g. flow cytometry
    • G01N2015/1402—Data analysis by thresholding or gating operations performed on the acquired signals or stored data
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume or surface-area of porous materials
    • G01N15/10—Investigating individual particles
    • G01N15/14—Optical investigation techniques, e.g. flow cytometry
    • G01N2015/1486—Counting the particles
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/62—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light
    • G01N21/63—Systems in which the material investigated is excited whereby it emits light or causes a change in wavelength of the incident light optically excited
    • G01N21/64—Fluorescence; Phosphorescence
    • G01N2021/6417—Spectrofluorimetric devices

Definitions

  • the present invention relates to a method for quantification of particles emitting specific emitted entities (hereinafter “emittends”) and for characterization of the time-dependent behavior of the particles in a system comprising at least particles of a species j, in accordance with the preamble of claim 1 .
  • Such methods are conventionally employed in the field of fluorescence fluctuation spectroscopy, where particles emitting photons as emittends are quantified and characterized.
  • the particles are stimulated by means of an external light source, e.g. a laser, and the emission characteristics of the particles are determined by detection, whereby conclusions on the particles in the system can be drawn.
  • Fluorescence correlation spectroscopy has proven in the past to be a particularly advantageous method for quantifying and characterizing particles in a system, as described e.g. in EP 0 679 251 B1.
  • a system which conventionally is a solution comprising different particles having specific partial concentrations is measured by means of a confocal microscope lens.
  • a laser beam of a stimulating laser is imaged into the system with such a confocal microscope lens in such a way that just a very small stimulation volume is illuminated by the laser, on the other hand the photons being emitted by the particles present in the stimulation volume are imaged by the confocal microscope lens on a detector.
  • the stimulation volume can thus be restricted to less than 1 fl by the known confocal microscope lenses.
  • FCS Fluorescence Signals are determined which indicates over the measuring period the number of photons the detector detects during the measuring period.
  • the time-dependent course of the number of detected photons can be read from the fluorescence signal.
  • Information on the diffusion constant of the emitting particles, the emission characteristics of the particles and the partial concentration of the particles can then be ascertained from the time-dependent autocorrelation function of the fluorescence signal.
  • FCS methods are also known from prior art, in which the measuring period is divided into a plurality of time intervals having equal length and the number of detected photons is determined for each of the time intervals. From this the fluorescence signal is determined which represents the time-dependent course of the number of photons detected during the time intervals over the measuring period. Accordingly this allows obtaining information on the partial concentrations of the particles and the diffusion constants of the different species of particles in the system from the autocorrelation function of the fluorescence signal.
  • the FCS method is based on the determination of the time-dependent performance of the fluorescence signal by means of the autocorrelation function.
  • information on particles of different species in the monitored system is determined from this time-dependent data.
  • the brightness i.e. the absolute number of detected photons
  • Obtaining information on the particles of the system is carried out by means of a set of data which is significantly reduced compared to the data collected by the detector, which comprises inter alia the absolute number of detected photons.
  • the FCS method is barely suited for several fields of application.
  • the FCS method is not suited for determining the partial concentrations of and characterizing particles of different species in the system, if the particles of different species exhibit similar mass and/or similar diffusion coefficients in the system.
  • the PCH (photon counting histogram) method is often used in combination with the FCS method.
  • the PCH method is based upon obtaining data by means of detection of photons which are emitted by particles in the system as is the FCS method, wherein the collection of data is carried out by means of a confocal microscope lens together with stimulation by a laser source, as described above.
  • the PCH method is e.g. described in Chen Y. et al., The photon counting histogram in fluorescence fluctuation spectroscopy, Biophysical Journal, 77, 553-567, 1999.
  • the number of photons which were detected in one time interval having a predetermined interval width is determined several times during the measuring period.
  • a photon count rate histogram is prepared from these data which indicates the distribution of the number of detected photons. Information on the absolute number of photons detected in the time interval is thus contained in the photon count rate histogram.
  • the PCH method is suited to allow quantifying of particles in a system comprising particles of different species, each having a different radiation characteristic, in particular determining the partial concentrations of the different species.
  • the PCH method is not suited for analyzing time-dependent behavior of the particles in the system, since the data determined in the PCH methods do not contain time-dependent information.
  • the FCS method as well as the PCH method allows determining data characterizing the particles of different species in the system under measurement by evaluating the data determined by the FCS or PCH method, respectively, by means of numerical algorithms.
  • the PCH and FCS methods offer access to different typical data which characterize the particles of different species and the partial concentrations thereof in the system. It is always necessary to carry out the FCS method and the PCH method completely in order to obtain an comprehensive characterization. This is time-consuming and requires significant computer resources.
  • the combination of PCH and FCS methods cannot guarantee complete analysis of a system having particles of different species. In particular the analysis of the time-dependent behavior in a system having different species which exhibit similar diffusion constants and/or similar masses is hardly possible.
  • a method having the features of claim 1 is proposed to solve said technical object.
  • the emissions of particles during a measuring period are detected in a measurement step.
  • the number n of the emissions having been detected in a time interval of predetermined interval with within the period is thereafter ascertained and stored.
  • the evaluation can be carried out for several time intervals having the same interval width within the measuring period.
  • the time intervals can in particular be chosen so that they do not overlap.
  • a distribution function p(n) of the number of detected emissions n is determined.
  • the distribution function p(n) indicates the relative frequency of determining the different values for the number n.
  • the method according to the invention is distinguished by stipulating different bin times ⁇ as interval width, carrying out the evaluation for each bin time ⁇ and determining a distribution function p ⁇ (n).
  • the distribution function p ⁇ (n) indicates the relative frequency of detecting the different values for the number n of emissions in a time interval having the bin time ⁇ as interval width.
  • a distribution function p ⁇ (n) is determined for each bin time ⁇ .
  • a distribution function p ⁇ (n) is determined.
  • the moments m i, ⁇ Mess are determined as characteristics of the distribution function p ⁇ (n) in a conventional manner.
  • moment functions m i Mess ( ⁇ ) dependent on bin time are derived wherein the moment functions m i Mess ( ⁇ ) are formed across the individual points m i, ⁇ Mess at the respective bin times ⁇ .
  • the first moment function m ⁇ Mess ( ⁇ ) is prepared using the first moments m i, ⁇ Mess of the distribution function p ⁇ (n) at the respective bin times ⁇ . This applies accordingly to higher moment functions relating to higher moments, which are prepared from corresponding higher moments of the distribution functions p ⁇ (n) for different bin times ⁇ .
  • the analysis of the measurement data determined within the measuring period or the emissions determined within the measuring period, respectively, is carried out on the basis of the measurement data set, which comprises the moment functions m i Mess ( ⁇ ).
  • the evaluation of the data set comprises a numerical fit of a theoretical signal function comprising moments m i sig ( ⁇ ) of a theoretical signal distribution P sig (n, ⁇ ) to a measurement function comprising the moment function m i Mess ( ⁇ ), whereby constants characterizing the particles in the system and contained in P sig (n, ⁇ ) are ascertained.
  • Measurement function and theoretical signal function comprise m i Mess ( ⁇ ) or m i sig ( ⁇ ), respectively, in the sense that the measurement function can be represented by a function comprising m i Mess ( ⁇ ) and the signal function can be represented by a function comprising m i sig ( ⁇ ).
  • the measurement data set can comprise the time resolved progress of the number n during the measuring period as well as the p ⁇ (n) calculated therefrom.
  • the measurement data set comprises such extensive information on the measurement results, that an extensive characterizing and quantifying of the particles in the system is possible.
  • the evaluation can be based on a theoretical signal distribution P sig (n, ⁇ ) of the system, which is defined so as to comprise those parameters which are independent on bin time ⁇ .
  • P sig (n, ⁇ ) of the system, which is defined so as to comprise those parameters which are independent on bin time ⁇ .
  • the parameter ⁇ 0,j represents a characteristic detection brightness of a particle of species j. Therefore the parameter ⁇ 0,j is a constant comprising characteristic properties of a particle of the species j as well as of the measurement device. E.g. the constant ⁇ 0,j comprises properties of the particle of the species j like the cross-section or the quantum efficiency of such a particle. Besides of this ⁇ 0,j comprises device-dependent values like e.g.
  • the function ⁇ ( ⁇ right arrow over (r) ⁇ ) represents the local dependency of the local detection rate.
  • the local detection rate changes depending on where the particle is located in the system to be measured.
  • the function ⁇ ( ⁇ right arrow over (r) ⁇ ) can e.g. comprise the locally dependent properties of the lens system employed in the measuring apparatus and/or the locally dependent properties of the stimulation profile, e.g. the locally dependent intensity distribution of the imaging of a laser into the system.
  • the function ⁇ ( ⁇ right arrow over (r) ⁇ ) can e.g. be given via the normalized point-spread-function of the measuring apparatus by the equation
  • R represents the volume in which a particle of the species j can be located theoretically during the measurement.
  • the formulation of the normalized point-spread-function for a measuring apparatus is sufficiently known and can be carried out by the skilled person for the measuring apparatus used in each case.
  • each particle of the species j has a specific local probability distribution and a specific local emission probability.
  • the local probability distribution represents the probability for the particle being in a specific location.
  • the local emission probability represents the probability that the particle emits an emittend.
  • the skilled person may select a suitable known probability distribution as the expected specific local probability distribution and as the expected specific local emission probability of the particles of one species and is able to accept it for formulating the theoretical signal distribution P sig (n, ⁇ ).
  • the particles have to be located in a specific measuring volume V in order to enable detection of emissions of the particles.
  • the measuring volume V is determined in the system by means of the imaging volume of a stimulating laser, if the emission of the particles is based upon stimulation by a laser and only particles stimulated by a laser can emit emittends.
  • the measuring volume V can also be defined by a limitation of the volume given by the measuring apparatus. Such limitation can e.g. be a purely geometrical outer confinement of the system like a wall. Such limitation can e.g. also be predetermined on the basis of the lens which projects the emission of the particles onto the detector.
  • the theoretical signal distribution P sig (n, ⁇ ), the definition of which is based upon the assumptions mentioned above, comprises as parameters which do not depend on the bin time the local detection rate ⁇ j ( ⁇ right arrow over (r) ⁇ ), a particle concentration c j and a decay time ⁇ j of the particle species j.
  • the theoretical signal distribution P sig (n, ⁇ ) of the system comprises the noise performance of the measuring apparatus.
  • the noise performance of the measuring apparatus can assumed to be constant with time or can also be assumed as dependent from time or variable with time, respectively, in particular as a value having a statistic distribution.
  • the noise performance of the measuring apparatus can be neglected.
  • the so defined theoretical signal distribution P sig (n, ⁇ ) also comprises the noise performance of the measuring apparatus.
  • the theoretical signal distribution P sig (n, ⁇ ) of the system comprises corresponding parameters independent of the bin time for each of the particle species.
  • the signal distribution P sig (n, ⁇ ) moments m i sig ( ⁇ ) are ascertained in a conventional manner.
  • the first moment m i sig ( ⁇ ) and the second moment m i sig ( ⁇ ) of the signal distribution P sig (n, ⁇ ) can be ascertained.
  • the theoretical signal function is defined which comprises at least one of the ascertained moments m i sig ( ⁇ ) as well as a measurement value function which comprises at least one of the ascertained moment functions m i Mess ( ⁇ ). At least constants with respect to at least the parameters ⁇ 0,j and ⁇ j are then ascertained by means of a numerical fit of the theoretical signal function with the measurement value function.
  • the theoretical signal function can exhibit the same K-th moments m K sig ( ⁇ ) as correspondingly has the measurement value function over m K Mess ( ⁇ ).
  • the theoretical signal function can e.g. encompass m 1 sig ( ⁇ ), m 2 sig ( ⁇ ), and m 3 sig ( ⁇ ) and the measurement value function can encompass m 1 Mess ( ⁇ ), m 2 Mess ( ⁇ ), and m 3 Mess ( ⁇ ).
  • the theoretical signal function and the measurement value function can only exhibit the same K-th moments at each time.
  • the theoretical signal function exhibits only such K-th moments m i sig ( ⁇ ) which are correspondingly contained as m K Mess in the measurement value function and vice versa.
  • the K-th moments m K sig ( ⁇ ) of the theoretical signal function can have the same functional relationship to one another as have the corresponding K-th moments m K Mess ( ⁇ ) in the measuring value function.
  • the method according to the invention is suited for an analysis of any system which comprises particles of at least one species emitting specific emittends.
  • emittends can e.g. be photons, but e.g. can also be ⁇ -particles or gamma radiation.
  • emittends can also be photons, but e.g. can also be ⁇ -particles or gamma radiation.
  • gamma radiation Depending on the nature of the emittends and the properties of the particles stimulation from outside can be necessary or not.
  • the method according to the invention is based upon the fundamental finding that the moment functions m i Mess ( ⁇ ) depending on bin time comprise information on the absolute number of detected emissions as well as on the time-dependent performance of the particles.
  • the measuring value function comprising moment functions m i Mess ( ⁇ ) a function is thus provided which is based upon measurement values and which allows an extended and detailed analysis of the system or the particles in the system, respectively.
  • the invention suggests varying the bin time ⁇ , determining the distribution function p ⁇ (n) for each bin time ⁇ and determining the moment functions m i Mess ( ⁇ ) depending on bin time.
  • the method according to the invention is based upon the finding that by numerically fitting the measurement value function to a theoretical signal function comprising moments m i sig ( ⁇ ) of the theoretical signal distribution P sig (n, ⁇ ) constants can be ascertained which do not depend on bin time and which characterize the particles in the system and the concentration of the particles in the system.
  • the constants relate at least to the parameters ⁇ 0,j and ⁇ j , in particular also c j , in particular c j of the different species, if there is a system having several species.
  • the constants can e.g. be identical to ⁇ 0,j , c j and/or ⁇ j . E.g.
  • the constants can contain ⁇ 0,j , c j and/or ⁇ j in one constant in each case, they can e.g. have a mathematical ratio with constant values, e.g. numbers.
  • ⁇ j can encompass several components, which have to be determined independent from each other by means of numerical fitting, e.g. a separate decay component ⁇ j,x , ⁇ j,y , ⁇ j,z for each spatial dimension.
  • the theoretical signal function for each of the decay components can encompass a constant relating to the respective decay component.
  • the parameters not depending on bin time are determined by numerical comparison of the theoretical signal function with the measurement value function.
  • the numerical comparison is carried out computer-controlled, wherein the constants contained in the theoretical signal function which encompass the parameters not depending on bin time, i.e. ⁇ 0,j and ⁇ j and in particular c j , are adjusted in numerical comparison so that the functional course of the theoretical signal function adjusts to the functional course of the measurement value function.
  • the skilled person is thus able to define a theoretical signal distribution P sig (n, ⁇ ) taking in account the aforementioned assumptions in order to determine the theoretical signal function in view of a compromise which does not request too much computer capacity and still allows a sufficiently precise specification of the relevant constants, in particular of parameters ⁇ 0,j , c j and ⁇ j .
  • the specific local probability distribution indicates the probability to find a particle of species j at a certain time t in a certain location ⁇ right arrow over ( ⁇ ) ⁇ .
  • the specific local probability distribution can be indicated by the function ⁇ ( ⁇ right arrow over (r) ⁇ ,t).
  • the skilled person can determine the function ⁇ ( ⁇ right arrow over (r) ⁇ ,t) by means of the boundary conditions of the measuring apparatus and the measuring environment.
  • the spatial limitation of the system only and/or the transportation of the particles or the movement of the particles in the system, e.g. whether there is a free stochastic diffusion or fluidic transport, respectively, may be relevant.
  • the local probability distribution can e.g. be defined by means of the known homogeneous transport equation, wherein under the assumption that the particle is in location ⁇ right arrow over (r) ⁇ 0 at time t 0 for ⁇ ( ⁇ right arrow over (r) ⁇ ,t
  • the skilled person can stipulate corresponding local probability distributions relating to other measuring apparatuses having different measuring environments, which predetermine different boundary data for the determination of ⁇ ( ⁇ right arrow over (r) ⁇ ,t
  • the nature of the emission process of the particles is relevant for assuming a specific local emission probability.
  • the emission process can depend e.g. on the internal structure of the particles and/or on interaction of the particles with the stimulating field.
  • the internal structure can e.g. be a related to the emission generating transitions within the particles.
  • the specific local emission probability can e.g. often be approximated by a Poisson distribution.
  • the skilled person can make use of known local emission probabilities depending on the nature of the emittends in order to define the theoretical signal distribution P sig (n, ⁇ ).
  • the present invention is also based upon the finding that it is possible to define a theoretical signal distribution P sig (n, ⁇ ) by defining a specific local probability distribution which determines the probability to find a particle in place ⁇ right arrow over (r) ⁇ at time t, by determining a specific local emission probabilities which indicates the probability for a particle in location ⁇ right arrow over ( ⁇ ) ⁇ to emit an emittend, by determining a specific local detection rate ⁇ j ( ⁇ right arrow over (r) ⁇ ) taking in account specific brightness characteristics of the particle and detection characteristics of the measuring apparatus, and by assuming a specific measuring volume V in which the particle has to be located in order to enable detecting emissions of the particle.
  • E j (n,t) is the specific local emission probability of the particles of the species j and wherein the decay time ⁇ ⁇ is taken in account when formulating ⁇ j (r,t).
  • ⁇ ⁇ can e.g. indicate an average reaction time during which a particle of the species j can react so that it can no longer emit emittends.
  • ⁇ ⁇ can e.g. indicate the travelling time of a particle of species j through the measuring volume V.
  • ⁇ j can e.g. indicate the half-life for the decay of a particle of species j.
  • the noise performance of the detector unit can e.g. be taken into account by:
  • the noise performance of the measuring apparatus can be considered by R(n,t).
  • the signal distribution for a system having an arbitrary number m of particles can e.g. be formulated by an m-fold convolution of P sig 1 (n, ⁇ ) and under consideration of the probability to find m particles in the measuring volume as being:
  • the detector noise can e.g. be considered only now by:
  • the local detection rate ⁇ j ( ⁇ right arrow over (r) ⁇ ), the specific local probability distribution and the specific local emission probability are separately established for each species in order to define the theoretical signal distribution P sig (n, ⁇ ).
  • P sig the theoretical signal distribution
  • the specific local probability distribution and the specific local emission probability can be stipulated as being identical for each species.
  • an individual theoretical species signal distribution P sig 1 (n, ⁇ ) particular for each species of particles is determined in applying the method according to the invention to a system having s different species of particles.
  • the theoretical signal distribution P sig (n ⁇ ) of the system is then determined by s+1-fold convolution of the s different signal distributions P sig j (n, ⁇ ) of the s different species and at noise signal distribution P noise (n, ⁇ ).
  • the noise signal distribution P noise (n, ⁇ ) can be neglected, so that the theoretical signal distribution P sig (n, ⁇ ) of the system can be defined by s-fold convolution of the s different signal distributions P sig j (n, ⁇ ).
  • Said advantageous embodiment allows a comparatively simple setup of theoretical species signal distributions P sig j (n, ⁇ ) separately for each individual species. Thereafter a concrete indication of the theoretical signal distribution P sig (n, ⁇ ) of the total system encompassing the particles of the s different species can be carried out by convolution of the s theoretical species signal distributions.
  • the measuring volume V utilized for defining P sig (n, ⁇ ) is being defined by means of a bin time dependent effective volume V eff,j ( ⁇ ) introduced fictitiously.
  • the effective volume V eff,j ( ⁇ ) is not the volume spatially delimited by the measuring apparatus. Rather the effective volume V eff,j ( ⁇ ) is a bin time dependent means introduced by a mental experiment in order to simplify defining the theoretical signal distribution P sig (n, ⁇ ). Introducing the effective volume comes along with the assumption that the probability has the value 1 for finding a particle the emission of which is detected during the bin time ⁇ in the volume defined by the fictitiously introduced effective volume ⁇ V eff,j ( ⁇ ).
  • Introduction of the effective volume allows normalization to ⁇ V eff,j ( ⁇ ) of functions building P sig (n, ⁇ ) or P sig (n, ⁇ ), respectively, as e.g. the signal distribution for a single particle.
  • the consideration of the definition of ⁇ V eff,j ( ⁇ ) and the corresponding normalization of P sig (n, ⁇ ) is a particularly advantageous embodiment, since the corresponding local integration can be carried out spatially unlimited without creating divergent expressions, whereby integration is simplified, P sig (n, ⁇ ) can be expressed simpler so that the numerical fit can be carried out simpler and more exactly.
  • the bin time dependent mean value n( ⁇ ) and the bin time dependent variance ⁇ 2 ( ⁇ ) are determined for each bin time ⁇ from the distribution functions p ⁇ (n), wherein the measuring value function is defined so as to include n ( ⁇ ) and ⁇ 2 ( ⁇ ).
  • the measuring value function is defined so as to include n ( ⁇ ) and ⁇ 2 ( ⁇ ).
  • the skilled person understands this as equivalent with determining the first moment m 1, ⁇ Mess and the second moment m 2, ⁇ Mess of the distribution function p ⁇ (n) for each bin time ⁇ and then constituting the bin time dependent moment functions m i, ⁇ Mess ( ⁇ ) and m 2 Mess ( ⁇ ), since n ( ⁇ ) and ⁇ 2 ( ⁇ ) can be calculated unambiguously from m i Mess ( ⁇ ) and m 2 Mess ( ⁇ ) and vice versa.
  • the measurement value function comprises functions of 2 which can be converted unambiguously into at least one m i Mess ( ⁇ ) or into a function composed of several m i Mess ( ⁇ ). According to the invention also such functions can be determined from P ⁇ ( ⁇ ) instead of m i Mess ( ⁇ ).
  • the measurement value function so as to comprise n ( ⁇ ) and ⁇ 2 ( ⁇ ) the expression of the measurement value function can be kept simple. Still it is guaranteed that the sufficiently precise determination of the relevant parameters is possible by means of numerical fit between the measurement value function and the theoretical signal function. This is based upon the finding that in n ( ⁇ ) and ⁇ 2 ( ⁇ ) there is sufficient information for guaranteeing sufficiently exact quantification and characterizing of the particle in the system from the measurement data obtained during the measuring period.
  • m 1 sig ( ⁇ ) represents the first moment and m 2 sig ( ⁇ ) represents the second moment of P sig (n, ⁇ ).
  • ⁇ 2 ( ⁇ ) and the mean value can be determined from m 1 Mess ( ⁇ ) and m 2 Mess ( ⁇ ).
  • This advantageous embodiment is based upon the finding that the numerical fit between the correspondingly defined measurement value function and the correspondingly defined theoretical signal function can be carried out particularly simple in a plurality of applications, that a very precise indication of the bin time independent parameters ⁇ 0,j , c j and ⁇ j is possible by the corresponding numerical fit with relative low effort, and that the evaluation of the obtained measurement data is possible with limited effort because of the limitation to considering the first and second moments of the theoretical signal function and the distribution functions p ⁇ (n).
  • the numerical fit takes place based upon the measuring value function Q( ⁇ ), wherein the numerical fit is carried out by means of the relation
  • Such a numerical fit aims at fitting of the measurement detection rate with the detection rate expected from theory by introducing as divisor into the fit.
  • This embodiment may enable carrying out a particularly simple numerical fit allowing determination of the constants with little calculating effort.
  • such fit can be specifically simplified by carrying out the fit with a limit value consideration at
  • the theoretical signal function is being defined as time-dependent by means of carrying out several measurements, each at a specific point in time assigned to the respective measurement.
  • a bin time dependent measurement value function is determined for each of the measurements assigned to the respective specific point in time by approximating each measurement value function with an approximation graph, for example by numerical or graphical fit. From the approximation graph of each of the measurement value functions the value of the respective measurement value function is determined, which this measurement value function has for
  • the numerical fit is carried out by fitting the limit value of one of the measurement functions to the limit value of the theoretical signal function at a point in time assigned to this measurement value function.
  • measurements are carried out at different specific points in time and the measurement value function is determined for each measurement.
  • the time span of the measuring period of each measurement is significantly lower than the time difference between one measurement and the measurements adjacent in time to this one.
  • the measurement period for each one of the measurements can extend over the same time span.
  • the time span between measurements can be constant for all measurements.
  • the time span of the measuring period can be less than 10%, in particular less than 1% of the time distance of one measurement to the measurement or measurements next to this one.
  • the time span of 5 seconds can be established for the measuring period of one measurement and the time distance to the next measurement to 5 minutes.
  • the start of the measuring period of the assigned measurement is determined by a specific point in time.
  • data sets are generated at each specific point in time by the explained variation of bin time and the measurement value function is determined from these data sets for the specific point in time.
  • the measurement value function is determined from these data sets for the specific point in time.
  • the numerical fit can particularly simply be carried out by assuming that some of the constants from the theoretical signal function are constant over the time and others are variable with time. E.g. the concentrations of the particles in the system can be assumed as being variable with time and the brightness in emission of emittends by the particles can be assumed as constant with time. Numerical fit can be carried out particularly simply in the embodiment described by stipulating the measurement value function to be
  • a mean local detection rate ⁇ 1,j of the species j is established by means of the integral
  • ⁇ _ 1 , j ⁇ R ⁇ ⁇ ⁇ 3 ⁇ r ⁇ ⁇ ⁇ j ⁇ ( r ⁇ )
  • the constant ⁇ 1,j referring to the parameter ⁇ 0,j can thus be ascertained in the numerical fit between the theoretical signal function and the measuring value function.
  • the mean local detection rate ⁇ 1,j corresponds per definition to the detection rate of a hypothetical particle of species j, which can be determined by means of the detector used in the measurement; said particle of species j emits emittends at any point in time during the measurement with its particle specific characteristic rate summarized over the spatial volume.
  • the invention is based upon the finding that by introducing the mean local detection rate ⁇ 1,j the formulation of the theoretical signal distribution ⁇ 1,j can be highly simplified. This allows for a simpler numerical fit between measuring value function and theoretical signal function.
  • a corresponding mean local detection rate ⁇ 1,s can be established for each of the s species.
  • the advantageous embodiment is based upon the approach of simplifying the expression for the theoretical signal distribution by replacing the location dependent function ⁇ j ( ⁇ ) with a spatially summarizing consideration by a constant ⁇ 1,j on bin time which can be determined by numerical fit in accordance with the method of the invention and comprises the constant ⁇ 0,j .
  • a further advantageous embodiment is characterized in that in applying the method according to the invention to the stochastic transport of the particles for defining the theoretical signal distribution P sig (n, ⁇ ) shifts and mean detected number of emissions of an individual particle of the species j is established to be
  • ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 0 ⁇ ⁇ ⁇ ⁇ t ⁇ ⁇ R ⁇ ⁇ ⁇ 3 ⁇ r 0 ⁇ ( ⁇ R ⁇ ⁇ ⁇ 3 ⁇ r ⁇ ⁇ ⁇ j ⁇ ( r ⁇ ) ⁇ ⁇ j ⁇ ( r ⁇ ⁇ , t
  • R can represent a distinct space, e.g. a limited measuring volume or e.g. an unlimited space, in each case depending on the approach in defining the theoretical signal distribution P sig (n, ⁇ ).
  • integration over an unlimited space can grant advantages, since the integrals are simpler executable analytically.
  • integration can be carried out over an unlimited space if the bin time dependent fictitious effective volume as explained above is introduced for defining P sig (n, ⁇ ) for the measuring volume V being used and is considered correspondingly in the definition.
  • the theoretical signal distribution P sig (n, ⁇ ) can immediately be formulated in a simple manner assuming a specific distribution of the emission events for particles of species j.
  • the distribution to be selected for that purpose depends on the nature of the particles of species j and on the emittends to be emitted.
  • a binomial distribution, a Gaussian distribution or a Poisson distribution can e.g. be assumed depending on the application area of the method according to the invention.
  • n 0, 1, . . . .
  • the method according to the invention is here applied to a system having s different particle species and the noise performance of the measuring apparatus is considered by means of the noise constant ⁇ .
  • the numerical fit can be carried out immediately and simply, if an appropriate ⁇ 1,j ( ⁇ ) for the particles of species j for each of the s different particle species is being inserted.
  • the present embodiment is based upon, and to carry out the integration over the unlimited space in analogy to the determination of ⁇ 1,j .
  • characteristic properties of the particles like the diffusion constant of particles or the decay time of particles and the concentration of one particle species can be determined simply and precisely by numerical fit.
  • ⁇ j ⁇ ( r ⁇ ) ⁇ 0 , j ⁇ exp ⁇ ( - 2 a xy 2 ⁇ ( x 2 + y 2 ) ) ⁇ exp ⁇ ( - 2 a z 2 ⁇ z 2 )
  • ⁇ j ( ⁇ ) is assumed as local detection rate.
  • ⁇ j ( ⁇ ) is represented by a Gaussian function.
  • the equation given for ⁇ j ( ⁇ ) is one option to represent ⁇ j ( ⁇ ) for the definition of the theoretical signal distribution P sig (n, ⁇ ).
  • x, y and z are the local coordinates of the spatial vector.
  • a corresponding local dependency of ⁇ j ( ⁇ ) is e.g. an approximation yielding very good results for the method according to the invention in case of applying the method according to the invention to a system where the particles to be analyzed emit photons as emittends and are stimulated to emit by 1-photon stimulation.
  • P sig (n, ⁇ ) Depending on the area of application other local dependencies can also be assumed for defining P sig (n, ⁇ ). E.g. corresponding known dependencies of the emission can be applied in the case of applying the method to a system with particles emitting photons and being stimulated by two-photon-stimulation and for applying the method to a measurement with STED microscopy.
  • a mean detected emission number of a single particle of species j is stipulated to be
  • ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 2 ⁇ 2 ⁇ ⁇ 0 , j 2 ⁇ ⁇ a ⁇ ⁇ j 2 ⁇ ⁇ ⁇ - ⁇ ⁇ ⁇ ⁇ ⁇ z 0 ⁇ ( erf ⁇ [ 1 ⁇ j ⁇ ( z 0 v + ⁇ ) ] - erf ⁇ [ z 0 v ⁇ ⁇ ⁇ j ] ) 2
  • ⁇ j a 2 ⁇ ⁇ v
  • ⁇ j indicates the time reduced by ⁇ square root over (2) ⁇ in which one particle travels along distance a in the direction of transport.
  • the decay time ⁇ j can be construed as being a time which characterizes the time span during which one particle of species j is present within the volume where the particle is able to emit emittends in such a manner that they can be detected by a detector.
  • Distance a can be determined e.g. by the lens system of the measuring apparatus which projects the emittends from the measuring volume to the detector.
  • ⁇ j allows characterizing the transport properties of particles of species j in the system.
  • the mean detected emission number of a single particle z of the species j is stipulated to be
  • ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 0 , j 2 ⁇ ⁇ j ⁇ ( 1 - 1 1 + ⁇ ⁇ j )
  • the diffusion transport can e.g. be construed as an isotropic decay of particles which are able to emit emittends during the measurement before the decay and can no longer emit emittends after the decay.
  • the diffusion transport can be construed as a mass transport by diffusion of particles wherein emittends from the particles can be detected only when they are located in a specific measurement volume.
  • ⁇ j can e.g. indicate the time for decay of particles of species j.
  • ⁇ j can characterize the time needed by particles of species j to travel across the measurement volume.
  • ⁇ j can e.g. be described by
  • ⁇ j a 2 8 ⁇ ⁇ D j
  • the mean detected emission number of a single particle z of the species j can be stipulated to be
  • ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 0 , j 2 3 2 ⁇ ⁇ 0 ⁇ ⁇ ⁇ t ( 1 + t ⁇ j , xy ) ⁇ 1 + t ⁇ j , z
  • ⁇ j,xy and ⁇ j,z are decay times for particles of species j which are included in decay time ⁇ j .
  • the diffusion transport can be interpreted differently depending on the application of the method. If the diffusion transport is construed as a real mass transport of the particles in the system to be analyzed as explained above, ⁇ j,xy and ⁇ j,z can be described by
  • FIG. 1 is the distribution of individual molecules assumed for determination of P sig (n, ⁇ ) in an illustrative embodiment.
  • FIG. 2 is the theoretical signal distribution P sig (n, ⁇ ) as function of bin time for different values of n as determined according to one embodiment.
  • FIG. 3 is the schematic layout of a measuring apparatus as schematic diagram.
  • FIG. 4 is a graphic representation of the result of the numerical fit according to one embodiment of the invention.
  • FIG. 5 is a graphic representation of the numerical fit according to another embodiment of the invention.
  • FIG. 6 is a graphic representation of the results of the numerical fit of still another embodiment of the invention.
  • FIG. 7 is a graphic representation of the characteristic data obtained from the numerical fit in the embodiment according to FIG. 6 .
  • FIG. 8 is a graphic representation of the results of the numerical fit according to still another embodiment of the invention.
  • FIG. 9 is a graphic representation of the characteristic data obtained from the fit according to FIG. 8 .
  • FIG. 10 is a graphic representation of simulations of the behavior of parameters of a system to be investigated.
  • FIG. 11 is the graphic representation of measurements relating to the behavior of parameters of the system shown in FIG. 10 .
  • ⁇ ⁇ 1 , j ⁇ ( ⁇ ) n - 1 ⁇ ⁇ ⁇ - ⁇ 1 , j ⁇ ( ⁇ ) ( ⁇ R 3 ⁇ ⁇ 3 ⁇ r 0 ⁇ ⁇ j ⁇ ( r ⁇ ) ) ⁇ ( 1 - ⁇ - ⁇ 1 , j ⁇ ( ⁇ ) ) ⁇ 1 , j ⁇ ( ⁇ ) .
  • ⁇ j ⁇ ( m , ⁇ j ⁇ ( ⁇ ) ) ⁇ j m ⁇ ( ⁇ ) m ! ⁇ ⁇ - ⁇ j ⁇ ( ⁇ ) .
  • P 1,j (n, ⁇ ) indicates the probability to detect n emissions from one individual particle of species j being located in the fictitious volume ⁇ V eff,j ( ⁇ ) within a bin time ⁇ .
  • a corresponding P 1,j (n, ⁇ ) can for example be determined in the same way as in the illustrative embodiment shown.
  • other mathematical transformations and simplifications are also possible, however, in order to obtain an expression for P 1,j (n, ⁇ ) as simple as possible and still as exact as possible.
  • the theoretical signal distribution P sig (n, ⁇ ) is being determined by means of determining P 1,j (n, ⁇ ) as an auxiliary value.
  • P sig (n, ⁇ ) is defined through P sig (n, ⁇ ) by means of the assumption that in the system observed m particles being able to emit emittends during the measurement are present in the volume V eff,j ( ⁇ ) out of which the particles can emit detectable emittends.
  • the distribution of the number of emissions detected from the m particles can be represented by the m-fold convolution
  • ⁇ j ⁇ ( m , ⁇ j ⁇ ( ⁇ ) ) ⁇ j m ⁇ ( ⁇ ) m ! ⁇ ⁇ ⁇ - ⁇ j ⁇ ( ⁇ )
  • P sig (n, ⁇ ) represents the sum of the combinatorial possibilities to detect n emittends, wherein the contribution of the signal distribution for emittends emitted by just one particle is weighted with the corresponding product of the occupation number in each case.
  • m l,j 1 is the l-th moment of the signal distribution for a single particle of species j.
  • the moments m k,j 1 can be calculated by means of the characteristic function ⁇ (t) of the signal distribution of a single particle of species j, P 1,j (n, ⁇ ), wherein the conventional definition of ⁇ (t) and conventional mathematical transformations can be applied.
  • ⁇ ⁇ ( t ) ⁇ _ 1 , j V eff , j ⁇ ( ⁇ ) ⁇ ⁇ - ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 1 , j ⁇ ( ⁇ ) ⁇ ( ⁇ ⁇ 1 , j ⁇ ( ⁇ ) ⁇ ⁇ ⁇ ⁇ ⁇ t - 1 ) .
  • S(k,l) represent constant coefficients, that are Stirling numbers of second kind as occurring frequently with combinatorial problems. They describe the number of different options for partitioning a permutation of k elements into l cycles.
  • m 2 sig c j ⁇ 1,j ⁇ (1+ c j ⁇ 1,j ⁇ + ⁇ 1,j ( ⁇ ).
  • mean value n and variance ⁇ 2 can be calculated by
  • the mean value of the signal distribution corresponds to the rate of a single particle of species j summed up over the whole volume and multiplied with bin time ⁇ and particle density c j .
  • the variance deviates from the expected value by a factor of 1+ ⁇ 1,j , so that the signal distribution is not a Poisson distribution.
  • the theoretical signal function can e.g. be determined from n sig and ⁇ sig 2 by means of the definition of the theoretical signal function through Q sig with
  • P sig (n, ⁇ ) as well as the moments m i Mess ( ⁇ ) can be determined under the assumptions according to the invention, as explained with the illustrative embodiment shown and a theoretical signal function for the numerical fit can be formulated. In the illustrative embodiment shown above this has been carried out by way of example for a system where just one species of emitting particles j is expected. Of course other mathematical transformations and intermediate assumptions or intermediate definitions, respectively, are possible in order to arrive at an expression for P sig (n, ⁇ ) as well as m i sig ( ⁇ ).
  • the definition in accordance with the illustrative embodiment for a system having only one particle species j as mentioned above can be applied in an analogous manner for defining P sig (n, ⁇ ) for a system having s different particle species.
  • P sig (n, ⁇ ) for a system having s different particle species.
  • Several methods of calculation can be applied for this purpose.
  • the particles of the different species can diffuse within the system freely and unaffectedly, and that the number n of emittends detected within a time period is composed of the contributions n 1 originating from particles of the s different species.
  • n can be represented by
  • n noise takes in account detection events originating from noise of the measuring apparatus.
  • Noise can e.g. be detector noise and/or noise induced by a source of stimulation.
  • a probability according to the species signal distribution for the species l P sig l (n, ⁇ ) can be indicated for each number n l .
  • P sig l (n, ⁇ ) Provided stochastic independency one can obtain the mix probability of the system having s different particle species can be shown by means of an s-fold convolution of the s different signal distributions P sig l (n, ⁇ ) as:
  • P sig ( n , ⁇ ) ( P sig 1 ⁇ circle around (x) ⁇ . . . ⁇ circle around (x) ⁇ P sig s ⁇ circle around (x) ⁇ P noise )( n , ⁇ ).
  • P sig (n, ⁇ ) can be indicated in the present illustrative embodiment as explained in the first illustrative embodiment relating to a system having only one particle species j.
  • P sig (n, ⁇ ) can be indicated in the present illustrative embodiment for a system having s different species by:
  • P 1,l (v, ⁇ ) indicates the probability for detecting v emissions of a single particle of species l during bin time ⁇ .
  • ⁇ 1 ( ⁇ ) and ⁇ 2 ( ⁇ ), respectively indicate the mean population number for the species 1 and 2, respectively, and P i,1 and P i,2 , respectively, represent the probability distribution for the number of emissions during the bin time for a particle of species 1 or species 2, respectively.
  • the noise signal distribution can be taken into account by means of the convolution of the noise signal distribution with the signal distribution P sig of the system having s different species as explained above. Assuming a Poisson noise, as is often the case and therefore is conventionally assumed when approximating noise performance, while the noise signal distribution is given by
  • the moments m i sig ( ⁇ ) can be determined in analogy to the calculation for a system having just one species j depicted above from this recursive formula for P sig (n, ⁇ ) relating to the theoretical signal distribution for a system having s different species and taking into account the noise signal distribution P noise (n, ⁇ ) assuming a Poisson noise distribution.
  • the moments m i sig can be indicated as:
  • This theoretical signal function can thus be employed immediately in the numerical fit of a corresponding measurement value function in order to characterize a system having s species of particles.
  • this illustrative embodiment is based upon the assumption that the noise of the measuring apparatus can be expressed as Poisson noise having a rate of ⁇ .
  • Such an illustrative embodiment can yield very well approximated results of the constants to be determined, in particular if the signal noise ratio is high.
  • the measurement for detecting the emissions of the particles in the system to be monitored is carried out by means of a measuring apparatus as depicted in FIG. 3 .
  • the illustrative embodiment as described deals with quantifying and characterizing particles in the system emitting photons as emittends.
  • a light source 2 is needed for stimulating the particles in the system. Particles emit photons after being stimulated, wherein the emission events are distributed statistically.
  • the present measuring apparatus is for 1-photon stimulation.
  • a laser is being used as light source 2 in the measuring apparatus.
  • the light emitted by light source 2 is focused by an illumination lens 3 , filtered by a stimulation filter 4 and deflected to a lens 6 by the dichroic mirror 5 .
  • the lens 6 focuses the light emitted by light source 2 into the sample plane 100 .
  • Sample 1 comprising the system to be analyzed is located in the focus of lens 6 of the measuring apparatus so that the light emitted by light source 2 is focused in sample 1 comprising the system to its smallest volume in the measuring apparatus.
  • the volume to which the light emitted by light source 2 is focused within a sample 1 is 0.5 fl.
  • Particles in sample 1 are stimulated to emit photons by light from light source 2 .
  • Photons emitted by particles in sample 1 are focused by lens 6 and reach tube lens 8 through dichroic mirror 5 and an emission filter 7 removing laser light from the beam.
  • Tube lens 8 concentrates the beam formed by the emitted photons.
  • the beam reaches detection unit 10 through an extremely small pinhole 9 .
  • the measuring apparatus described is configured for allowing detection of single photons emitted by particles in sample 1 .
  • the data sampling unit 11 connected to the detection unit 10 ensures time resolved detection. This means that data sampling unit 11 records at which point in time photons in the detection unit were detected in each case.
  • Computer 12 allows evaluation of the detections of photons detected by data sampling unit scaled with time.
  • the detection unit 10 comprises a counter suited for detecting individual photons. In the illustrative embodiment described here avalanche photodiodes are employed for this purpose. When detecting a photon the detection unit 10 provides an output pulse to the data sampling unit 11 which is there provided with a timestamp and stored in a data storage medium in the computer 12 . In the illustrative embodiment described the data sampling unit is a FPGA board allowing the time resolved storage of the photon detections.
  • the measurement is carried out during the measuring period T.
  • the detected emissions are stored in the computer 12 during the whole measuring period T in time resolved manner. Different interval widths are established thereafter for evaluation.
  • the measuring period T is divided in
  • m i Mess ⁇ ( ⁇ ) ⁇ n ⁇ n i ⁇ p ⁇ ⁇ ( n ) .
  • FIGS. 4 and 5 show the results of measurement and evaluation for a system having only particles of one species. Measurement points of two different measurements are shown in FIG. 4 .
  • the chemical composition of the system was identical in each case for both measurements.
  • the velocity of the fluidic mass transport the particles in the system are subject to is different for both measurements.
  • the measurement values of ⁇ ( ⁇ ) determined by the measurement are drawn as measurement points.
  • the square measurement points result from the first measurement during which a fluidic transport velocity v 1 predominated in the system.
  • the measurement points with circular shape result from a second measurement during which a fluidic transport velocity v 2 predominated in the system.
  • the functions ⁇ ( ⁇ ) were formed in each case by means of the measurement values of the first and the second measurements, respectively.
  • the solid lines represent the theoretical signal functions after having determined the constants of the theoretical signal function by the numerical fit between the above-mentioned relation. From FIG. 4 it is evident that the course of the theoretical signal function is in very good agreement with the course of the measurement values for ⁇ ( ⁇ ).
  • ⁇ j ⁇ ( r ⁇ ) ⁇ 0 , j ⁇ ⁇ - 2 a 2 ⁇ r ⁇ 2 .
  • ⁇ 1,j ( ⁇ ) in the present case of deterministic fluidic transport results as:
  • ⁇ j a 2 ⁇ v .
  • the concentration c j of the species j in the system is not a constant comprised in the theoretical signal function when applying the method according to the invention to a system with only one emitting species j.
  • the only constants to be determined in the numerical fit are ⁇ 0 , v and a, wherein ⁇ j can be expressed as dependent on a and v as explained above.
  • the numerical fit can be simplified by a limit value consideration. This limit value consideration
  • the measuring value functions for both measurements are approaching the same limit value
  • the method according to the invention can be applied to a system having particles of different species in a similar way as in the preceding illustrative embodiment for fluidic mass transport in the system having just one emitting particle species j.
  • measurement values from five different measurements as well as the model theoretical signal function as determined by numerical fit are represented graphically in FIG. 5 for the corresponding five different measurements.
  • the detection rate ⁇ ( ⁇ ) is shown as measurement value function at single measurement points of the corresponding measurements in FIG. 5 , too.
  • the particles were subject in each case to the same velocity of the fluidic transport.
  • the system analyzed contains particles of one or two out of two species predetermined for all measurements.
  • the system differs in its composition or in the concentrations of particle species, respectively, between the respective measurements.
  • Each of the particle species has a different characteristic detection brightness ⁇ 0,j .
  • the theoretical signal distribution for the numerical fit according to FIG. 5 was formed by the function
  • FIGS. 6 to 9 Applications of the method according to the invention for applying to systems having particles subject to a diffusion transport in the system are described in FIGS. 6 to 9 by means of further illustrative embodiments.
  • the same mathematical transformations and definitions are the exemplary start point for the illustrative embodiments shown in each case.
  • a corresponding numerical fit between measurement value function and signal function can be evidently carried out as well by different mathematical approaches.
  • ⁇ right arrow over (r) ⁇ 0 ,t 0 ) has also to be determined as in the examples relating to FIG. 4 and FIG. 5 .
  • This probability distribution was approximated by means of Dirac's delta function in the case of a fluidic transport in FIGS. 4 and 5 . This approximation is not applicable in case of diffusion transport. Instead the probability distribution ⁇ ( ⁇ right arrow over (r) ⁇ ,t
  • ⁇ ⁇ ( r ⁇ , t ⁇ r ⁇ 0 , t 0 ) ( 4 ⁇ ⁇ ⁇ ⁇ D ⁇ ( t - t 0 ) ) - 3 2 ⁇ ⁇ exp ⁇ [ - ( r ⁇ - r ⁇ 0 ) 4 ⁇ D ⁇ ( t - t 0 ) ] .
  • a spherical measurement symmetry that is spherical symmetry of the spatial dependency of the detection rate in the measurement volume, can be assumed in an illustrative embodiment not shown graphically here.
  • a system having only one emitting species, which emits photons, is being analyzed.
  • the local dependency of the detection rate ⁇ ( ⁇ right arrow over (r) ⁇ ) is being assumed as
  • ⁇ ⁇ ( r ⁇ ) ⁇ 0 , j ⁇ ⁇ - 2 a 2 ⁇ r ⁇ 2
  • ⁇ 1,j ( ⁇ ) can thus be indicated similarly by means of the local probability distribution ⁇ j ( ⁇ right arrow over (r) ⁇ ,t
  • ⁇ ⁇ ( ⁇ ) ⁇ 1 , j ⁇ ( ⁇ ) ⁇ .
  • a spheroidal [rotation ellipsoidal] symmetry of the spatial dependency of the detection rate in the measurement volume is assumed.
  • the local detection rate ⁇ j ( ⁇ ) has to be based upon another local dependency.
  • the system is being stimulated by 1-photon-stimulation and the local dependency can be taken into account by a Gaussian function according to
  • ⁇ j ⁇ ( r ⁇ ) ⁇ 0 , j ⁇ exp ⁇ ( - 2 a xy 2 ⁇ ( x 2 + y 2 ) ) ⁇ exp ⁇ ( - 2 a z 2 ⁇ z 2 ) .
  • ⁇ 1,j can be indicated to be:
  • ⁇ 1 , j ⁇ ( ⁇ ) ⁇ 0 , j 2 3 2 ⁇ ⁇ z ⁇ xy - ⁇ z ⁇ 2 ⁇ ⁇ ⁇ xy ⁇ ( arc ⁇ ⁇ tan ⁇ [ ⁇ z ⁇ xy - ⁇ z ⁇ 1 + ⁇ ⁇ z ] - arc ⁇ ⁇ tan ⁇ [ ⁇ z ⁇ xy - ⁇ z ] ) .
  • ⁇ describes a parameter of the apparatus.
  • ⁇ ⁇ ( ⁇ ) ⁇ 1 , j ⁇ ( ⁇ ) ⁇ ,
  • FIG. 6 a the measurement value function ⁇ ( ⁇ ) resulting from the measurement data as described above is represented for individual measurement points. It is evident from FIG. 6 a that the measurement value function ⁇ ( ⁇ ) increases rapidly at small T. This unusual course of ⁇ ( ⁇ ) is based upon the performance of the detection unit 10 as for example noise or dead time of the detector. The measuring value function of the detector noise alone has to be determined and then has to be subtracted from the measurement value function ⁇ ( ⁇ ) of the measurement data according to FIG. 6 a in order to correct for this effect.
  • FIG. 6 b shows the measurement value function ⁇ ( ⁇ ) adjusted for detector noise. In the method described in accordance with FIGS.
  • detector noise is determined as an intermediate step and subtracted from the measured measurement value function ⁇ ( ⁇ ) before carrying out the numerical fit so as to generate an adjusted measurement value function ⁇ ( ⁇ ).
  • This adjusted measurement value function ⁇ ( ⁇ ) is then utilized for the numerical fit with the theoretical signal function and is then employed as described above in applying the method to a system with particles subject to diffusion transport.
  • the measurement value function is adjusted by subtracting the noise of the measuring apparatus.
  • noise of the measuring apparatus can be neglected in defining the theoretical signal function while precise results can at the same time be obtained by numerical fit and simple numerical fit is possible due to the simple formulation of the theoretical signal function.
  • FIG. 6 c The theoretical signal function after determination of the constants thereof is depicted in FIG. 6 c .
  • the constants have been determined by numerical fit assuming a rotation ellipsoidal symmetry in the illustrative embodiment according to FIG. 6 and FIG. 7 and the numerical fit is carried out by means of the relation:
  • ⁇ ⁇ ( ⁇ ) ⁇ 0 , j 2 3 2 ⁇ ⁇ 2 - 1 ⁇ ⁇ ⁇ z ⁇ ⁇ ( ln ⁇ ( ⁇ + 1 ⁇ - 1 ) - ln ⁇ ( ⁇ ⁇ 1 + ⁇ ⁇ z + 1 ⁇ ⁇ 1 + ⁇ ⁇ z - 1 ) ) .
  • FIG. 6 d the superposition of the measurement value function ⁇ ( ⁇ ) with the theoretical signal function (as shown alone in FIG. 6 c and resulting from the numerical fit). From FIG. 6 d it is evident that the method according to invention allows for an extraordinary agreement of the theoretical signal function with the measurement value function ⁇ ( ⁇ ). This is based upon the assumptions according to the invention allowing formulation of the theoretical signal function and the measurement value function in such a way that a simple and thus also precise numerical fit requiring low characterization effort is possible. The assumptions and transformations explained above contribute in particular to this result when being applied to the illustrative embodiment described in FIGS. 6 and 7 .
  • FIG. 7 depicts an illustrative embodiment of the method according to the invention in which a measurement and a numerical fit were carried out for characterizing a system accommodating emitting particles of only one species j similar to the illustrative embodiment according to FIG. 6 .
  • the results of a total of 12 measurements are represented in FIG. 7 .
  • a first concentration c 1 of the species in the system in the fourth to sixth measurement a concentration c 2 , in the seventh to ninth measurement a concentration c 3 and in the tenth to twelfth measurement a concentration c 4 was set in preparing the system.
  • Values for the characteristic detection brightness of the particles of species j in the system are indicated in FIG. 7 a for all 12 measurements.
  • the value of the characteristic detection brightness ⁇ 0,j of species j does not change dependent on the set concentration of the species in the system.
  • the decrease of the value for ⁇ 0 during each measurement series comprising three tests for each concentration c 1 , c 2 , c 3 and c 4 is due to the so-called “bleaching” of the particles during the measurement series causing a decrease of the characteristic brightness of the particles.
  • the characteristic detection brightness ⁇ 0 increases in the course of the measurement series. From FIG. 7 a it is evident that the method according to the invention provides reproducible results in regard of the characteristic detection brightness of the particles.
  • FIG. 7 b indicates the apparatus parameter ⁇ which is calculated by means of
  • the apparatus parameter ⁇ is a constant for all concentrations of the species in the system, since in the present case ⁇ z and ⁇ xy are predetermined in such a way that ⁇ z and ⁇ xy characterize the travelling time across the measurement volume.
  • the particles of species j are not subject to any decay in the system but contribute as long to the detected rate as they are located within the measurement volume.
  • the fact that the apparatus parameter ⁇ is found to be constant by the numerical fit in the method according to the invention confirms the precision of the method according to the invention when analyzing systems having emitting particles.
  • the transversal decay time ⁇ xy for the 12 measurements is indicated in FIG. 7 c . It is evident from FIG. 7 c that irrespective of measuring errors a constant decay time ⁇ xy is determined for the particles of species j in the system.
  • the decay time ⁇ xy is to be construed as being the travelling time of the particles across the measurement volume in the xy-plane
  • FIG. 7 d indicates the relative concentrations c 1 :c 2 :c 3 :c 3 .
  • concentration of the particles in the first three measurements (c 1 ) is arbitrarily set to 1.
  • the relative concentrations of FIG. 7 d were determined by means of the relation of the mean detected number of photons n j of the measurements in accordance with the illustrative embodiment of the method according to the invention.
  • the relation n j c j ⁇ 1,j ⁇ was utilized.
  • ⁇ 1,j can be determined by means of the theoretical signal function, n j corresponds to the number of detected photons, so that c j can be determined for each measurement.
  • the set of measurement data utilized according to the invention comprises the time resolved recorded number n of detected emissions, these quantitative measurement values can be used in the evaluation so that the evaluation can be carried out comprehensively.
  • the relations of the concentrations can be determined by means of the relations of the mean occupancy numbers ⁇ j ( ⁇ ).
  • FIGS. 8 and 9 the results of another illustrative embodiment of the method according to the invention in an application to a system having one particle species j is shown.
  • FIG. 8 a depicts—similar to FIG. 6 a —the measured measurement value function ⁇ ( ⁇ ).
  • ⁇ ( ⁇ ) is adjusted for detector noise.
  • FIG. 8 c shows the theoretical signal function determined by the numerical fit with the adjusted measurement value function ⁇ ( ⁇ ) from FIG. 8 b .
  • FIG. 8 d the superposition of the graphs from FIGS. 8 b and 8 c is shown. It is evident from FIG. 8 d that an extraordinary fit of the theoretical signal function to the measurement value function for transmitting the constants characterizing the system and the particles, respectively, is possible.
  • ⁇ ⁇ ( ⁇ ) ⁇ 0 , j 2 3 2 ⁇ ⁇ 2 - 1 ⁇ ⁇ ⁇ z ⁇ ⁇ ( ln ⁇ ( ⁇ + 1 ⁇ - 1 ) - ln ⁇ ( ⁇ ⁇ 1 + ⁇ ⁇ z + 1 ⁇ ⁇ 1 + ⁇ ⁇ z - 1 ) )
  • FIG. 9 represents the results of the three measurements for the characteristic detection brightness ⁇ 0,j of species j
  • FIG. 9 b the results for the apparatus parameter ⁇
  • FIG. 9 c the results for the transversal decay time ⁇ xy
  • FIG. 9 d the relative concentrations having been determined relative to one another in the respective measurements.
  • the concentrations were determined as explained above relative to FIG. 7 , wherein c 3 , that is the concentration in the third measurement, was set to 1.
  • the determination of the values is carried out as explained with the example of FIGS. 6 and 7 similar to the illustrative embodiment according to FIG. 8 and FIG. 9 .
  • the standard fluorophore Alexa 488 was utilized as emitting particle species j when preparing the system in the illustrative embodiment according to FIGS. 6 and 7 .
  • the same measuring apparatus was used so that the same a xy can be assumed.
  • constants relating to the measuring apparatus can be determined by means of the calibration measurement, such as a xy or a 2 or also a in the case of spherical symmetry, which allows the numerical fit in the following evaluation to be carried out simpler and/or more precisely for a more precise determination of the constants characterizing the system or the particles, respectively.
  • FIGS. 10 and 11 The method according to the invention is explained by means of a further illustrative embodiment in FIGS. 10 and 11 .
  • the method is performed with a first system having only particles of species A and with the second system having only particles of species B as well as a third system having particles of species A, B, and AB.
  • Particle species A and B are particle species being educts emitting emittends and reacting to form product AB according to the reaction equation A+B AB with the kinetic constant (association constant) K ⁇ .
  • the product AB may again disintegrate into educts A and B with the rate constant (dissociation constant) K a .
  • Product AB emits the same light emitting emittends as educts A, B.
  • a local detection rate ⁇ A ( ⁇ right arrow over (r) ⁇ ), ⁇ B ( ⁇ right arrow over (r) ⁇ ), ⁇ AB ( ⁇ right arrow over (r) ⁇ ) and a mean local detection rate ⁇ A ( ⁇ right arrow over (r) ⁇ ), ⁇ 1,B , ⁇ 1,AB can be assigned to each of the particle species.
  • ⁇ eff is an efficiency factor taking in account the effects which restrict the emission of the product, as e.g. quenching effects.
  • the method according to the invention is very well suited for determining the partial concentrations or the time-dependent behavior of the respective partial concentrations c A c B and C AB , respectively, as outlined in the illustrative embodiment below.
  • the method according to the invention is suited for all particle species or mixtures of particle species, respectively, which can be represented as explained, e.g. also for mixtures of particle species comprising more than three different particle species.
  • antibodies GAR have been chosen as particle species A, antibodies RAM as particle species B and the complex of both antibodies as particle species AB.
  • Particle species A as well as particle species B were labeled with the fluorophore Alexa 488. Two fluorophores per particle are bound for each particle species. Thus the particles emit photons as emittends.
  • the partial concentrations c A , c B , and c AB as well as the association constant K ⁇ are to be determined by numerical fit between the theoretical signal function and measurement value function. Several measurements are performed in order to allow determination of the time dependency of the partial concentration values.
  • the theoretical signal function is defined time-dependent:
  • the measurement value function is determined for different specific points in time by performing a measurement assigned to the point in time at different specific points in time.
  • the measurement function utilized at each specific point in time t 0 is:
  • the number s of particle species is three which results from the particle species A, B, and AB.
  • the numerical fit between the measurement value functions assigned to the specific points in time and the theoretical signal function at the specific points in time is carried out by fitting the limit values of both functions at bin time ⁇ - 0 . It is necessary to form the limit value lim of the theoretical signal function. Assuming for simplicity negligible noise performance ⁇ , the limit is formed by
  • the particles of particle species A and the particles in particle species B are labeled with the same number of fluorophores and exhibit the same local detection rate.
  • the particle species AB has twice the number of fluorophores as compared to the particle species A and B so that the local detection rate of particle species AB is twice the detection rate of particle species A and B.
  • c A ⁇ ( t ) A 0 ⁇ ( A 0 - B 0 ) A 0 - B 0 ⁇ e ( B 0 - A 0 ) ⁇ K a ⁇ t .
  • ⁇ c B ⁇ ( t ) B 0 ⁇ ( A 0 - B 0 ) A 0 ⁇ e ( A 0 - B 0 ) ⁇ K a ⁇ t - B 0 .
  • ⁇ c AB ⁇ ( t ) A 0 ⁇ B 0 ⁇ e ( A 0 - B 0 ) ⁇ K a ⁇ t - 1 A 0 ⁇ e ( A 0 - B 0 ) ⁇ K a ⁇ t - B 0 .
  • FIG. 10 a shows the time-dependent behavior of the partial concentrations c A (t), c B (t), and c AB (t) which results by calculation from the rate equations mentioned above, wherein
  • the partial concentrations c A and c B are A 0 and B 0 , respectively, at the start of the reaction, that is at the point in time when particle species A is mixed with particle species B, and that the partial concentrations c A and c B decrease with time while the partial concentration of particle species AB, the particles of which are formed from particles A and B, increases.
  • the local detection rate of the mixture of the particle species A and B increases with increasing time.
  • the local detection rate starts at 50 kcps and ends at about 84 kcps. This is attributable to the effect that at complete reaction of particle species A with particle species B because of the higher number of particles of particle species B particles of particle species B are still present in the mixture, so that the local detection rate representing the mean local detection rate of one individual particle of the mixture and therefore takes in account the local detection rate of particles AB as well as that of particles B is not twice the starting rate.
  • Each measurement value was determined by performing one measurement during one measuring period at one measurement point in time t 0 followed by determining the measurement value function by varying the bin time and then determining the limit value of the measurement value function for
  • the limit values of the measurement value functions for the different systems were determined, that is for a first system having particle species A, a second system having particle species B and a third system where the particle species A and particle species B are mixed, so that there particle species AB occurs. While no reaction inhibiting substance was added to the third system in the measurement according to FIG. 11 a , urea was added as reaction inhibiting substance in the measurement according to FIG. 11 b . It is evident from FIGS. 11 a , 11 b that the limit values of the measurement value function for the first and the second system remain approximately constant independent on the measurement point in time, while the limit value of the measurement value function for the third system in the case of FIG.
  • the method according to the invention can be employed in any measuring environment.
  • the method according to invention is suited for biochemical analysis in solutions or on surfaces. Surfaces can be functionalized units or can also be natural units as e.g. cell membranes.
  • the method according to the invention is suited for use in fluidic systems. E.g. measurements using a construction being designed as a Y-structure type, where two different particle species are fed through both legs of the Y which are mixed at the merger point so that by measuring at the merger point and with increasing distance from this point the performance of the mixture can be ascertained.

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