Systems and methods to quantify the transition rates of a stochastic system having a plurality of states

Abstract

The present invention relates to the study of stochastic systems and, more specifically, to systems and methods to quantify the transition rates in stochastic systems having a plurality of states. An aspect of the invention relates to a method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, comprising the steps of: providing the stochastic system; generating an equivalent electric circuit that represents a probability current associated with the state transitions in the stochastic system; applying a periodic perturbation P to the stochastic system, having a driving frequency ω that is varied within a predetermined frequency range; measuring an impedance response of the stochastic system to the periodic perturbation, including at least an amplitude and a phase shift as functions of the driving frequency ω; mapping the values of the electrical component of the equivalent electric circuit such that its impedance Z matches the frequency-dependence of the amplitude and phase shift in the measured impedance response; and, quantifying the transition rates of the stochastic system based on the equivalent electric circuit.

Description

[0001]SYSTEMS AND METHODS TO QUANTIFY THE TRANSITION RATES OF A STOCHASTIC SYSTEM HAVING A PLURALITY OF STATES FIELD OF THE INVENTION The present invention relates to the study of stochastic systems and, more specifically, to systems and methods that utilize the concept of electrical impedance to characterize a stochastic system capable of existing in multiple states, and subsequently quantify the relevant transition rates. Stochastic systems, characterized by random or probabilistic behaviour, are prevalent in various fields of technology, including physics, chemistry, biology, and engineering. Identifying and quantifying the transition rates between different states in a stochastic system is essential for understanding its dynamics and predicting its behaviour. BACKGROUND In the study of stochastic systems, various models are considered to describe the transitions between different states within these systems. These transitions can occur due to various factors, such as molecular interactions, chemical reactions, or physical processes. However, in many applications, it is important to fully characterize the stochastic system by identifying all of the possible states that system can be in and accurately quantifying the transition rates between each pair of states. For instance, when considering as an example a DNA stem-loop molecule that can exist in either an open or closed state. In a solution containing multiple copies of this molecule, thermal fluctuations may cause individual molecules to randomly transition between these states, moving from open to closed and vice versa. As these transitions occur randomly in time, the dynamics of the system can be described as a stochastic process. By accurately determining the transition rates of the molecule, it is possible to assess the fraction of molecules in a specific state as a function of one or more process variables impacting the probability of transitions, such as temperature or concentration. Therefore, accurately quantifying the transition rates provides valuable insights into the underlying mechanisms and enables the prediction and control of system behaviour. One approach for determining the transition rates in a stochastic system involves studying the stationary state properties of the system. By carefully measuring and manipulating the parameters affecting the stationary state, the ratio of the transition rates can be quantified. However, this method requires precise knowledge of the possible states and the properties of the molecules or processes involved in the transitions—information that may not be available for newly discovered or unidentified stochastic systems. Moreover, this approach may not capture real-time dynamics relevant, particularly in systems with higher complexity with more than two states or multiple interacting components. B. Cleuren et al., in Physica A 552 (2020) 122789, introduced the concept of impedance in a stochastic context to study the steady-state distribution of a simple stochastic system. However, this concept is not directly applicable for quantifying transition rates in stochastic systems. Therefore, while existing methods have provided some insights into stochastic systems, there remains a need for improved approaches that offer more accurate and efficient quantification of transition rates. Additionally, there is a need for methods that can extract more detailed information about the system, such as entropic and energetic characteristics, and / or identify hidden states within the stochastic system. SUMMARY OF THE INVENTION The present invention addresses the aforementioned needs by providing a novel approach to determine the probability of state transitions in a stochastic system. The present invention builds upon the understanding that transition rates can be manipulated through a time-dependent perturbation, effectively driving the stochastic system out of its stationary state. By controlling the parameters of the perturbation and observing the system's response to the applied perturbation, information can be extracted to comprehensively characterize the stochastic system. The information extraction employs the concept of electrical impedance applied in an equivalent electrical circuit to characterize the stochastic system's properties. By adapting the properties of the periodic perturbation, such as temperature or concentration, the present invention can be applied to a broad spectrum of stochastic systems in various fields of technology. Therefore, by way of the systems and methods described herein, the limitations of conventional techniques can be overcome by directly linking the measured response to the underlying transition probabilities, enabling a more real-time and dynamic assessment of the transition rates that provides more accurate predictions of system behaviour and moreover allows for increased depth of information extraction, including entropic and energetic information, or to identify hidden states of the system. A first overview of various components of the invention according to the present disclosure is given hereinbelow, after which specific embodiments will be described in more detail. This first overview is meant to aid the reader in understanding the technological concepts more quickly, but it is not meant to identify the most important or essential features thereof, nor is it meant to limit the scope of the present invention. An aspect of the present invention relates to a method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, the method comprising the steps of: - providing the stochastic system; - generating an equivalent electric circuit that represents or models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit comprises a plurality of electrical components; wherein the equivalent electric circuit has an impedance ^ related to the transition rates of the stochastic system; - applying a periodic perturbation ^ to the stochastic system, wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; - measuring an impedance response of the stochastic system to the periodic perturbation in a stationary time-periodic state, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the electrical component of the equivalent electric circuit such that its impedance ^ matches the frequency-dependence of the amplitude and phase shift in the measured impedance response; - quantifying the transition rates of the stochastic system based on the equivalent electric circuit. Another aspect of the present invention relates to a computer-implemented method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, wherein the computer-implemented method is operable on a processor of a computing device, the method comprising the steps of: - generating an equivalent electric circuit that represents or virtually models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system; - receiving sensor data indicative of an impedance response of the stochastic system to a periodic perturbation; wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; wherein the impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; - quantifying the transition rates of the stochastic system based on the equivalent electric circuit; and, - optionally, outputting the quantified transition rates of the stochastic system. Another aspect of the present invention relates to a system for quantifying a plurality of transitions rates of a stochastic system that can be in a plurality of states, the system comprising: - a perturbator means configured for applying a periodic perturbation to the stochastic system, wherein the periodic perturbation has a driving frequency ω that can be varied over a predetermined frequency range; - a sensor means configured for measuring an impedance response of the stochastic system to the periodic perturbation, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - optionally, a receiver means adapted for receiving the stochastic system, wherein the perturbator means and the sensor means are disposed on the receiver means; - a processing unit operatively or communicatively connected to the perturbator means and sensor means, wherein the processing unit is configured to: - generate an equivalent electric circuit that represents or models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system; - map the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; and, - quantify the transition rates of the stochastic system based on the equivalent electric circuit. In an embodiment the equivalent electric circuit includes, preferably consist of, a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system. In an embodiment the values of the resistor and the capacitor of each parallel branch are mapped such that the impedance Z of the equivalent electric circuit matches the frequency-dependence of the amplitude and phase shift in the measured impedance response. In an embodiment each parallel branch includes, preferably consists of, a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system. In an embodiment the stochastic system is a two-state stochastic system, comprising a single component that can be in a ground state S0 and a first, higher energy state S1, wherein the impedance response of the stochastic system to the periodic perturbation is measured for at least one concentration of the component; wherein the transition rates are quantified by -- determining the ratio A of the ground state S0 and the first state S1 rate; -- determining a characteristic frequency ^, defined as the frequency for which the phase shift equals 45 degrees; - calculating the transition rate R from the first state S1 to the ground state S0 based on R = and / or - calculating the transition rate from the ground state S0 to the first state S1 based on R . ^. In the above embodiment the impedance response of the stochastic system to the periodic perturbation is measured for at least one concentration of the single component. In the above embodiment the characteristic frequency is determined by plotting the amplitude and the phase shift as functions of the driving frequency ω, and selecting the frequency for which the phase shift substantially equals 45 degrees, for example 40 degrees to 50 degrees, preferably exactly 45 degrees. In another embodiment the stochastic system is a two-state stochastic system, comprising a plurality of components that can interact with each other to form a ground state S0 and a first, higher energy state S1; wherein the impedance response of the stochastic system to the periodic perturbation is measured for at least two different concentrations of at least one of the components, ^^and ^^; wherein the transition rates are quantified by - determining a characteristic frequency for each of the at least two different concentrations, ^^and ^^; - calculating the transition rate R from the first state S1 to the ground state S0 based on ^ - calculating the ratio A of the ground state S0 and the first state S1 rate based on ^ =^^^^^^^^ , and / or - calculating the transition rate from the ground state S0 to the first state S1 based on R . ^. In the above embodiment the stochastic system comprises a plurality of components, the components include at least one probe component and at least one target component, wherein the probe is configured to interact with the target component to form one of the states of the stochastic system; and perturbation is measured for at least two different concentrations of the target component, ^^and ^^. In the above embodiment the characteristic frequency is determined by plotting the amplitude and the phase shift as functions of the driving frequency ω, and identifying a crossover point in the phase shift plot; preferably wherein the crossover point is characterized by a sudden change or discontinuity in the slope of the phase response curve. In an embodiment the periodic perturbation comprises one or more types of perturbations selected from the group consisting of harmonic perturbations, step perturbations, pulse perturbations, periodic perturbations with non-harmonic waveforms; preferably step perturbations . In an embodiment the periodic perturbation is applied by varying at least one parameter selected from the group consisting of temperature, pressure, concentration, electromagnetic radiation preferably comprising one or more wavelengths within the visible, ultraviolet, and / or infrared spectrum, and / or a combination thereof, to induce the impedance response in the stochastic system; preferably temperature. In an embodiment the periodic perturbation is applied in discrete steps within the predetermined frequency range, with each step maintaining a constant driving frequency for a predefined duration before transitioning to the next driving frequency; preferably wherein the constant driving frequency is maintained until the stochastic system reaches a stationary time periodic state. In an embodiment the stochastic system comprises a probe molecule and a target molecule, wherein the probe molecule is configured to interact with the target molecule to form a complex, and wherein the method further comprises characterising the kinetic and thermodynamic properties of the stochastic system based on the quantified transition rates. In an embodiment the stochastic system comprises deoxyribonucleic acid (DNA), and wherein the method further comprises identifying mutations in a sequence of the DNA and / or distinguishing between different types of DNA based, on the quantified transition rates of the stochastic system. In an embodiment the stochastic system comprises a chemical reaction comprising a sequence of one or more intermediate reaction steps, and wherein the method further comprises identifying the presence of the one or more intermediate reaction steps based on the quantified transition rates of the stochastic system. An aspect of the present invention relates to a method for quantifying the change of enthalpy ΔH and / or the change in entropy ΔS of a stochastic system that can be in a plurality of states, comprising the steps of: - performing the method of any one of the preceding claims to quantify the transition rates of the stochastic system; wherein the stochastic system is provided at a reference temperature T^^^; wherein the applied periodical perturbation comprises a thermal perturbation based on T(^)= T^^^+ ^T(^), wherein ^^ is the perturbation strength around the reference temperature; - quantifying the change of enthalpy ΔH and / or the change in entropy ΔS for the stochastic system based on the quantified transition rates of the stochastic system and the reference temperature. In an embodiment the change of enthalpy ΔH and / or the change in entropy ΔS is quantified by - determining a maximum Amplitude û^^^based on the Amplitudes û in the measured impedance responses; - calculating the ratio ^ based on ^ =û^^^(^^^)^ ; - calculating ΔH based ^ - calculating ΔS based on ΔS = ^^ln(A)+ ^^T ^ , wherein ^^is the Boltzmann factor. DESCRIPTION OF THE FIGURES The following description of the figures relate to specific embodiments of the disclosure which are merely exemplary in nature and not intended to limit the present teachings, their application or uses. The following symbols and abbreviations are used throughout the figures and the present disclosure: state S; ground state S0 ; first state S1; transition rate R; transition rates ratio A; resistor ^; capacitor ^; Intensity I; time t; periodic perturbation ^; Temperature T; driving frequency ω; characteristic frequency ^; impedance ^; free energy ΔG; enthalpy difference ΔH; entropy difference ΔS; concentration c; Boltzmann factor ^^; Fluorophore F; Quencher Q. Figure 1 is a flow diagram of a method 100 for quantifying a plurality of transition rates of a stochastic system in accordance with aspects of the present disclosure. Figure 2 is a two-state stochastic system that can be in a (lower energy) ground state S0 and a (higher energy) first state S1. Figure 3 is a three-state stochastic system that can be in a ground state S0, a first state S1 and a second state S2. Figure 4 is an equivalent electrical circuit for the two-state stochastic system of Figure 1. Figure 5 is a DNA stem-loop molecule that can be in an open state S1 or a closed state S0. Figure 6 is a molecular beacon consisting of a Fluorophore(F)- and Quencher(Q)-conjugated DNA that can be in an open (fluorescent) first state S1 or a closed (non-fluorescent) ground state S0. Figure 7 shows the molecular beacon (fluorescence) intensity ^ as a function of time t for the molecular beacon of Figure 6, which is proportional to the probability P^by an optical amplification factor θ. Figure 8 is a single stranded DNA sequence (with no secondary structure) that can be utilized as a fluorophore (F) probe molecule to target a complementary single stranded DNA sequence. Figure 9 shows a generic configuration of a N-state stochastic system that can be extended to N- discrete states. Figure 10 is an equivalent electrical circuit for a N-state stochastic system of Figure 8. Figure 11 shows the occupation of the unbound (fluorescent) configuration in function of time for a simulation where an exemplary two-state model was perturbed at frequency ω = 0.6 and a period of about 10 sec. Detailed discussion of the system of Figure 11 can be found in Example 1. Figure 12 shows the response of an experimentally accessible signal to an applied temperature perturbation for the perturbed system of Figure 11. FIG.13 shows the frequency dependence of an amplitude response for the perturbed system of Figure 11. Figure 14 shows the frequency dependence of a phase shift response for the perturbed system of Figure 11. Figure 15 shows the frequency dependence of an amplitude response of a perturbed stochastic system consisting of pure wild-type and pure mutant DNA. Detailed discussion of the system of Figure 15 can be found in Example 2. Figure 16 shows the frequency dependence of a phase shift response of the perturbed system of Figure 15. Figure 17 shows the frequency dependence of an amplitude response of a perturbed system consisting of pure wild-type, pure mutant, and a mixture of wild-type and mutant DNA. Detailed discussion of the system of Figure 17 can be found in Example 3. Figure 18 shows the frequency dependence of a phase shift response of the perturbed system of Figure 17. Figure 19 shows the frequency dependence of an amplitude response of a three-state perturbed stochastic system compared to a two-state system. Detailed discussion of the system of Figure 19 can be found in Example 4. Figure 20 shows the frequency dependence of a phase shift response of the perturbed system of Figure 19. DETAILED DESCRIPTION In the following detailed description, the technology underlying the present disclosure will be described by means of different aspects thereof. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, and designed in a wide variety of different configurations, all of which are explicitly contemplated and make part of this disclosure. This description is meant to aid the reader in understanding the technological concepts more easily, but it is not meant to limit the scope of the present disclosure, which is limited only by the claims. As previously mentioned, the complete characterization of a stochastic system often requires identifying all possible states that the system can be in and determining the transition rates between each pair of states. The present invention builds upon the understanding that transition rates can be manipulated through a time-dependent perturbation, effectively driving the stochastic system out of its stationary state. By controlling the parameters of the perturbation and observing the system's response to the applied perturbation, information can be extracted to comprehensively characterize the stochastic system. The information extraction employs the concept of electrical impedance applied in an equivalent electrical circuit to characterize the stochastic system’s properties. This approach allows for the identification of any hidden states and the quantification of the relevant transition rates. The stochastic system can be any dynamical system that can be described by a discrete state Markov Process, where the future state of a system depends only on its current state and the probability of a transitioning to a particular state can therefore be described by transition state constants. In the subsequent description the stochastic system will be described by way of exemplary molecular species, single or multiple types, and the states will correspond to different configurations that molecules can be in. The transitions rates are those of the reversible reactions between the states. While these exemplary molecular species represent possible implementations of the invention, it should be noted that the method is in principle not limited solely to such embodiments. The principles and techniques outlined herein can be applied to a broad spectrum of fields of technology, such as (bio)chemical or physical processing and engineering, where random fluctuations and uncertainties play a significant role in shaping the system’s behaviour. Thus, the scope of the present invention extends to all such implementations and applications, as would be appreciated by those skilled in the art. An overview of various aspects of the technology of the present disclosure is given hereinbelow, after which specific embodiments will be described in more detail. This overview is meant to aid the reader in understanding the technological concepts more quickly, but it is not meant to identify the most important or essential features thereof, nor is it meant to limit the scope of the present disclosure. When describing specific embodiments, reference is made to the accompanying drawings, which are provided solely to aid in the understanding of the described embodiment. An aspect of the present invention relates to a method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, the method comprising the steps of: - providing the stochastic system; - generating an equivalent electric circuit that represents or models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit comprises a plurality of electrical components; wherein the equivalent electric circuit has an impedance ^ related to the transition rates of the stochastic system; - applying a periodic perturbation ^ to the stochastic system, wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; - measuring an impedance response of the stochastic system to the periodic perturbation in a stationary time-periodic state, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the electrical component of the equivalent electric circuit such that its impedance ^ matches the frequency-dependence of the amplitude and phase shift in the measured impedance response; - quantifying the transition rates of the stochastic system based on the equivalent electric circuit. Another aspect of the present invention relates to a computer-implemented method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, wherein the computer-implemented method is operable on a processor of a computing device, the method comprising the steps of: - generating an equivalent electric circuit that represents or virtually models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system; - receiving sensor data indicative of an impedance response of the stochastic system to a periodic perturbation; wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; wherein the impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; - quantifying the transition rates of the stochastic system based on the equivalent electric circuit; and, - optionally, outputting the quantified transition rates of the stochastic system. Another aspect of the present invention relates to a system for quantifying a plurality of transitions rates of a stochastic system that can be in a plurality of states, the system comprising: - a perturbator means configured for applying a periodic perturbation to the stochastic system, wherein the periodic perturbation has a driving frequency ω that can be varied over a predetermined frequency range; - a sensor means configured for measuring an impedance response of the stochastic system to the periodic perturbation, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - optionally, a receiver means adapted for receiving the stochastic system, wherein the perturbator means and the sensor means are disposed on the receiver means; - a processing unit operatively or communicatively connected to the perturbator means and sensor means, wherein the processing unit is configured to: - generate an equivalent electric circuit that represents or models a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system; - map the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; - quantify the transition rates of the stochastic system based on the equivalent electric circuit; and, - optionally, outputting the quantified transition rates of the stochastic system, preferably through an output means such as a display. It is understood, based on any of the embodiments described herein, that the method disclosed herein can be performed by the system described here. In other words, the hardware or software of the system can be adapted to perform the method and any of its steps, and vice versa, even if not explicitly described as such. For example, the system's processing unit may be adapted to execute a relevant step of the method when it receives a relevant signal or data as input. For example, this input could be in the form of sensor data acquired by one or more sensor means that are part of the system. The system’s processing unit may comprise a processor configured for performing the method in accordance with aspects of the present invention when provided with at least one relevant signal or data as input. FIG.1 is a flow diagram of a method 100 for quantifying a plurality of transition rates of a stochastic system in accordance with aspects of the present disclosure. It is understood that additional steps can be provided before, during, and after the steps of the method, and that some of the steps described can be replaced or eliminated for other implementations of the method. In the shown embodiment, the method 100 begins at step 101 by providing a stochastic system having a plurality of states. The stochastic system can be any dynamical system that exhibits random or probabilistic behaviour while transitioning between two or more states. As used herein, transition rates represent the frequency at which the stochastic system undergoes transitions from one state to another. The transition rates are related to the underlying processes that govern the system’s behaviour and evolution over time, which can be biological, physical or chemical in nature. The probability of a transition occurring within a specific time interval is proportional to the transition rate. Consequently, the rate equation describes the time evolution of the probabilities of different states in the system. In an embodiment the stochastic system may consist of a single component, such as a single type of molecule that can be in two or more states. Examples of such systems may include a DNA stem- loop molecule with open S1 and closed states S0, as shown in FIG.5. This embodiment has reduced complexity because knowledge of the concentration of the molecule is typically not necessary for the quantification of the transitions rates. In another embodiment the stochastic system may consist of multiple components, such as multiple types of molecules that interact with each other to form two or more states. Examples of such systems may include a DNA probe hybridizing to a single stranded DNA target present at a specific concentration c to form a complex. The DNA probe can be found in free state S1 (not bound to the target molecule) or in complex state S0 (bound to the target molecule), as shown in FIG.8. This embodiment has increased complexity because knowledge of the concentration is important for the quantification of the transitions rates. Returning to FIG.1, it is shown that at step 102 the method 100 can include generating an equivalent electric circuit representing a probability current associated with state transitions in the stochastic system. As mentioned above, the transition rates are related to the probability of a transition occurring within a specific time interval, but it does not account for the probability of finding a system in a specific state at a specific time. The probability current in this context refers to the rate of flow of probability between states due to these transitions. Accordingly, it represents the rate at which the probability of the system being in a specific state changes due to transitions to another state. Because the probability current is a vector, it can be represented as a current flowing in an electrical circuit. In an embodiment, the equivalent electrical circuit may be generated using a computer system. This process may include a step of providing the computer system with data related to the stochastic system and any other information relevant for the generation of the equivalent circuit. Advantageously, the equivalent electric circuit can be modelled as a virtual model that is interactable or adaptable by input of the user. The input can be provided through an input means, such as a keyboard, or as a set of instructions from another system, such as a modelling system. In an embodiment, the equivalent electric circuit may comprise a plurality of parallel branches, where the number of parallel branches equals the number of states in the stochastic system. For example, in a stochastic system having two states there can be two parallel branches, in a system having three states there can be three parallel branches, and so on. In an embodiment, each parallel branch may comprise an electrical component or combination of electrical components that represent the impedances of the stochastic system to a state transition. In a further embodiment, each parallel branch may include and preferably consist of a resistor and a capacitor connected in series, wherein the values of each resistor and capacitor pair are related to the transition rates in the stochastic system. Preferably, each parallel branch contains at most one resistor and / or at most one capacitor. The equivalent circuit advantageously does not contain any inductor or electrically complex component. Relevant to the aforementioned embodiments, it is important to appreciate that the configuration of the equivalent electric circuit is contingent upon the complexities of the stochastic system being perturbed. For instance, an equivalent electric circuit representing a stochastic system with two states will have reduced complexity compared to one representing a stochastic system with three or more states. Consequently, specific implementations of the method for a lower complexity stochastic system will be discussed further below, serving as a foundation for subsequently expanding to higher complexity stochastic systems. At step 103, the method can include applying a periodic perturbation with a variable driving frequency to the stochastic system. As used herein, the perturbation refers to an external influence, such as a force or input that affects the system’s state or behaviour over time. The perturbation is periodic, meaning that is applied at regular time intervals or frequencies. Advantageously, the driving frequency of the applied periodic perturbation may be selected to match the timescale of the stochastic system. Varying the driving frequency over a specific range can be achieved using techniques such as frequency sweeps or frequency modulation. In an embodiment the periodic perturbation can comprise one or more types of perturbations selected from the group consisting of harmonic perturbations, step perturbations, pulse perturbations, periodic perturbations with non-harmonic waveforms. Preferably, the periodic perturbation comprises step perturbations to vary the driving frequency over a range in a systematic manner (using frequency sweeps) as this lowers the complexity of the calculations. In an exemplary embodiment, the driving frequency ω can be at least 10-5Hz to at most 103 Hz, preferably at least 10-4Hz to at most 102 Hz. The driving frequency is advantageously adapted based on the properties of the stochastic system such that it can elicit a (sufficiently strong) impedance response from the stochastic system in order to effectively characterize its properties. In an embodiment the periodic perturbation can be applied by varying at least one parameter selected from the group consisting of temperature, pressure, concentration, electromagnetic radiation preferably comprising one or more wavelengths within the visible, ultraviolet, or infrared spectrum, and / or any other suitable perturbation, to induce a response in the stochastic system. The type of periodic perturbation is advantageously selected based on the properties of the stochastic system. In a further embodiment, wherein the periodic perturbation is a thermal perturbation, the perturbation can be applied by varying the perturbation strength around a reference temperature. In an exemplary embodiment, the perturbation strength can range from at least 0.1°C to at most 10.0°C, more preferably from 0.1°C to 9.0°C, more preferably still from 0.3°C to 8.0°C, more preferably still from 0.4°C to 7.0°C, more preferably still from 0.5°C to 6.0°C, more preferably still from 0.6°C to 5.0°C, more preferably still from 0.7°C to 4.0°C, more preferably still from 0.8°C to 3.0°C, more preferably still from 0.9°C to 2.0°C, more preferably still from 1.0°C to 1.5°C. In an embodiment the periodic perturbation can be applied by a perturbator means configured for applying a periodic perturbation to the stochastic system. The perturbator means may include one or more devices configured to vary a temperature, a pressure, a concentration, an electromagnetic radiation preferably comprising one or more wavelengths within the visible, ultraviolet, or infrared spectrum, and / or any other suitable perturbation, to induce a response in the stochastic system. The type of perturbator means is advantageously selected based on the properties of the stochastic system. In an embodiment the periodic perturbation can be applied continuously while transitioning through the predetermined frequency range. In another embodiment the periodic perturbation can be applied in discrete steps within the predetermined frequency range, with each step maintaining a constant driving frequency for a predefined duration before transitioning to the next driving frequency, preferably wherein the constant driving frequency is maintained at least until the stochastic system reaches a stationary time periodic state. In one embodiment, a plurality of periodic perturbations can be applied to the stochastic system, wherein one or more parameters of the stochastic system are adapted. Preferably, the adapted parameters of the stochastic system include temperature and / or concentration. This advantageous approach provides an additional experimental degree of freedom, enabling the extraction of additional information about the stochastic system. Building upon the aforementioned embodiment, it is also possible to adapt one or more parameters of the periodic perturbation when adjusting the parameters of the stochastic system. For instance, if the concentration of the stochastic system is increased, a different driving frequency range and / or intensity may be chosen for the periodic perturbation. This adjustment is made to elicit a specific impedance response from the stochastic system in order to effectively characterize its properties. In an embodiment, the periodic perturbation can be applied to a stochastic system in a stationary time-periodic state. This can allow investigating the dynamic behaviour of the stochastic to the applied perturbation. It should however be appreciated that the time-periodic state of the stochastic system is not relevant for quantifying of the transition rates when the periodic perturbation is applied, but rather when then the response is measured, as described below. At step 104, the method can include measuring an impedance response of the stochastic system to the periodic perturbation in a stationary time-periodic state. As described above, the applied periodic perturbation will change the stochastic system’s state or behaviour over time. The stochastic system, driven away from equilibrium by the applied perturbation, will gradually return to a stationary time-periodic state, which is a condition of the stochastic system where the probabilities of being in different states do not change with time because the rate of transitions from on state to another is exactly balanced by the rate of transitions in the reverse direction. This condition ensures that the probabilities are in equilibrium and therefore allow for a measuring a time independent response. As used herein, in reference to the stochastic system, impedance is a measure of the opposition that the system offers to the flow of the probability current. When the periodic perturbation is applied with a specific driving frequency, the impedance of the stochastic system may vary with the frequency of the perturbation. This frequency-dependent behaviour is referred to as the impedance response. In an embodiment, the impedance response may include an Amplitude and a phase shift of the system’s response at different frequencies. In an embodiment the response may include at least one of a fluorescence signal, an electromagnetic radiation signal preferably comprising one or more wavelengths within the visible, ultraviolet, or infrared spectrum, a surface plasma resonance signal, an electric signal, a change in conductivity, and / or a change in magnetic properties. The type of response is advantageously selected based on the properties of the stochastic system and the applied periodic perturbation. In an embodiment the response can be measured by a sensor means configured for measuring the response of the stochastic system to the periodic perturbation. The sensor means may include one or more sensor devices configured to measure a temperature, a pressure, a concentration, am electromagnetic radiation preferably comprising one or more wavelengths within the visible, ultraviolet, or infrared spectrum, and / or any other suitable sensor, to measure a response in the stochastic system. The type of sensor means is advantageously selected based on the properties of the stochastic system and the perturbator means. In an embodiment the response can be measured in real time. Alternatively, the response may be measured at certain times based on the frequency of applied periodic perturbation. The measurements may be stored in a memory within the sensor means or a control unit connected to the sensor means. At step 105, the method can include mapping the values of a plurality of components of the equivalent electric circuit such that its impedance corresponds to the measured impedance response. As previously described, by measuring the impedance response of the system as a function of the driving frequency, valuable information about the transition rates between different states of the stochastic system can be obtained. In an embodiment the measured impedance response can be analysed to determine a characteristic frequency that is frequency independent. Preferably, the characteristic frequency is determined by plotting the amplitude and the phase shift as functions of the driving frequency ω, and identifying a crossover point in the phase shift plot. The crossover point can be defined as a point in the slope of the phase response curve that shows by a sudden change or discontinuity in the slope of the phase response curve. Knowledge of the characteristic frequency can be used to map the values of a plurality of components of the equivalent electric circuit. However, as previously mentioned, because configuration of the equivalent electric circuit is contingent upon the complexity of the stochastic system being perturbed, so will be the values assigned to the electrical components present in the equivalent electric circuit. Consequently, specific implementations of the method for a lower complexity stochastic system will be discussed further below, serving as a foundation for subsequently expanding to higher complexity stochastic systems. At step 106, the method can include quantifying the transition rates of the stochastic system based on the component values of the equivalent circuit. As previously discussed, the values of the electrical components are related to the impedance associated with the corresponding state transition. Thus, once the component values in the equivalent circuit are determined, they can be used to quantify the transition rates between the corresponding states of the system. In an embodiment, the method can further include outputting the quantified transition rates and optionally any other parameters of the stochastic system as described herein. The outputting can be performed through an output means, such as a display, a printer, or as a set of instructions for another system. The output can be in the form of a report including at least the quantified transition rates, for example a digital report. In an embodiment, the quantifying of the transition rates may include identifying the presence of hidden states of the stochastic system. Regarding the latter, the present method has the advantage that if the values mapped to the plurality of components based on the measured impedance response do not complete the equivalent electrical circuit, it can be an indication that the stochastic system has another unknown state. In such a scenario, a step may be added that includes adapting the equivalent electrical circuit to include or more parallel branches that correspond to the one or more hidden states and recalculating the transition rates based on the adapted equivalent electrical circuit. In an embodiment, the stochastic system may consist of a single component that can be in two states. To further the understanding of the present invention, an overview of the theoretical background relevant for the generation of an equivalent electric circuit and quantification of the transition rates from said electric circuit for a single component stochastic system will be described in accordance with aspects of the present invention. In an embodiment, the stochastic system may consist of a single component system that can exist in two or more states. To provide a better understanding of the present invention, an overview of the theoretical background relevant to generating an equivalent electric circuit for a two-state stochastic system will be described. FIG.2 shows the simplest configuration of a single-component stochastic system, which can be in a ground state S0 and a first state S1. Suppose there are numerous independent copies of such a system, and due to thermal fluctuations, each individual copy can undergo transitions between these states, moving from the ground state S0 to the first state S1, and vice versa, from the first state S1 to the ground state S0. As these transitions occur randomly over time, the dynamics of the system can be described as a stochastic process. In such a scenario, the free energy of state ^ can be denoted as ^^, the probability of being in state ^ as ^^and the transition rates between the states by ^ and ^. ^. For a macroscopic system containing many independent copies, the probability ^^can be identified by the fraction of molecules in state ^. The time evolution of the probability distribution is governed by the Markov ( ^ ^ process, following the Master Equation: ^̇t)=^−^ ^ ^^ ^ ^^^ = ^ ^ ^ ^ ^^^, wherein the rates- ^^ −^ ^^containing matrix is called the Stochastic Matrix. Without any external interactions, starting from a random initial state, the stochastic system will evolve exponentially over time towards the stationary state of this Markov process, referred to as the equilibrium. It is given by the following formula: Hence, the equilibrium is uniquely determined by the ratio between on- and off-rate corresponding to ^. Further, statistical physics teaches that the equilibrium state is governed by the Boltzmann distribution ^ which provides the identification ^ = ^^^∆^, with the ^ difference in free energy being ∆^ = ^^− ^^> 0, inverse temperature ^ = and the Boltzmann factor ^^. The present invention is based on the fact that the rates in the above Master Equation can be perturbed in a time-dependent way, which drives the system inherently out of equilibrium. ^ = ^^^∆^shows that ^ is temperature dependent, which is sufficient for the invention to work, any other parameters capable of influencing the transition rates can also be used to perturb the system. For simplicity, the focus will be on temperature as the perturbing factor. A small time-dependent thermal perturbation δT(t). can be applied around a reference temperature ^^^^resulting in temperature being T(t)= ^^^^+ δT(t). This thermal perturbation prevents the systems from reaching an equilibrium state. It's worth noting that the rates ^ and ^. ^ generally depend on the temperature and become time dependent as well. In the long-time limit, when the solution to the Master Equation is applied to the system, the Master Equation can be rewritten using a time −^^ ^ δT t ^ − ^^ dependent stochastic matrix: ^ ^ + ^^ − ^^ ^ ^, in which ^ = an ^( )^^ + ^^ −^ ^ d ^^ − ^ ^^ ^ = ^^. Therefore, for a more general analysis, it can be assumed that both ^ and ^ are temperature dependent. As a consequence of this δT(t)-perturbation, the probabilities ^^to be in state 0 and ^^to be in state 1 will remain time-dependent, even after a long time. The same holds for probability currents between the different states, and the corresponding probability distribution ^(^)is referred to as the steady-state solution. Applying the above method to a two-state system results in the equivalent electric circuit shown in FIG.4, which illustrates the probability current between states 0 and 1 as an electrical current. The electric circuit consists of two parallel branches representing the two states of the system, the first branch consists of a resistor ^ ^ = ^^^, and the second branch consists of a resistor R^= ^^^and pacitor C =^^ ca^^^^^^connected in series. The impedance can be determined by the transition rates ^ ^ ^^ of the perturbed stochastic process through ^^^= 0, ^^^= and ^^= −^ (1 + ^). Consequently, when the frequency-dependence of the impedance is measured, the transition rates can be extracted. ^ The impedance of the equivalent electric circuit of FIG.4 is Z = R^−^^^. Using polar coordinates and switching to conductance σ, the inverse of impedance, this can be written as σ ^ and ^ = arctan ^ ^(^^^)^. Quantities that are directly accessible in experimental settings are often related to the properties and complexity of the stochastic system. In FIG.5, an example of a DNA stem-loop molecule is shown, which can exist in an open state S1 or a closed state S0. In a solution, there may be numerous independent copies of this molecule. Due to thermal fluctuations, individual molecules can undergo transitions between these states, transitioning from the open to the closed state, and vice versa. As these transitions occur randomly in time, the dynamics of the system can be described as a stochastic process. To experimentally track the state of the exemplary DNA stem-loop molecule shown in FIG.5, a molecular beacon can be used. FIG.6 illustrates an example of fluorophore(F)- and quencher(Q)- conjugated DNA. When the beacon is in the open state, the quencher and fluorophore are spatially separated, allowing the fluorophore to be accessible for optical readout. In the closed state S0, the quencher deactivates the fluorescence of the beacon. Consequently, the fluorescence of an ensemble of molecular beacons is proportional to the fraction of beacons in the open state S1 or, equivalently, to the probability for a beacon to be in the closed state S0. FIG.7 shows of the molecular beacon intensity ^(^), which is proportional to the probability P^by an optical amplification factor θ. Due to the thermal perturbation, the intensity exhibits harmonic fluctuations around its equilibrium value. This establishes a connection between the theoretical background described previously and an experimental implementation of the present invention. In a first step, the steady-state probability current can be determined ^ = σ. δT(t), and this quantity can be integrated to find the steady-state probabilities. When a harmonic perturbation δT(t)= δT ^^^^is used, the following can be derived: ^^⃗ (t)= Here ^ and ^ are provided according to the former equations, and relate to the amplitude and phase of the experimental signal of FIG.7, except for the optical proportionality constant θ. This proportionality constant can be removed by making the ^(^) measurement relative to its equilibrium value, more precisely, 1 − = 1^^^(^^) − ^ = ^ The left-hand side of the latter formula is directly accessible by experiment, and the right-hand side is directly related to the transition rates of the stochastic process. Hence, this equation allows quantifying transition rates between different states of a system in accordance with aspects of the present invention. An implementation of this formula according to aspects of the present disclosure is presented in Example 1 of the Examples section below. In another embodiment, the stochastic system may consist of a plurality of components that interact with each other to form two states. Advantageously, the plurality of interacting components collectively form the two states. As a straightforward extension to the previously discussed single component embodiment, another scenario can be considered where two or more molecules can bind to form a complex. In the example of DNA, FIG.8 shows a short single stranded DNA sequence (with no secondary structure) can be utilized as a fluorophore-probe (F) molecule. This probe serves the purpose of detecting the presence of a second molecule, the target (Q), which is another single-stranded sequence of DNA that is fully complementary to the sequence of the probe. When the probe binds to the target, it forms a double-stranded DNA complex that quenches the probe molecule’s fluorescence signal. Generally, this DNA complex is more stable, meaning it has a lower free energy, compared to the unbound state. Therefore, it will now be denoted as the ground state S0. In a situation where the target molecule has a DNA sequence that is not fully complementary to the probe, the complex can still possibly form but will have reduced stability. From this perspective, the probe serves as a sequence-specific sensor to detect the presence of the target. Lastly, it is assumed that the target is present at a certain concentration c. The temporal behaviour of this system now depends on the concentration c, and the transition rates discussed earlier (which were independent of c) need to be adapted. The transition from state S0 to state S1 is not affected by c, hence the off-rate ^^ remains c-independent. However, the on- rate is affected by concentration and becomes proportional to c, taking the form ^^. This suggests ^ a change of variables to ^^= ^^ and ^^= ^. With these adapted variables, it is found that ^^^^= ^^, and the functional form of the time evolution becomes identical to that of the former equation, namely ^(̇t) ^−^^^^^^^ ^^^^−^^^ ^^^^^. Therefore, all the above equations can be used without further adaptations. It can be appreciated that the inclusion of c -dependence provides an extra experimental degree of freedom. Concentration affects the on-rate, which, in turn, affects the speed of the reactions and the characteristic frequency of the system. This advantageous feature allows designing experiments within a frequency window that is experimentally more accessible. In another embodiment, the stochastic system may consist of a single components that can be in three or more states or a plurality of components that interact with each other to form three or more states. Expanding upon the previously discussed embodiments of a stochastic system that can be in two states, the theoretical background can be further developed to encompass higher complexity N-state stochastic systems, where N > 2, in order to generate a relevant equivalent electric circuit and quantify the relevant transition rates from said electric circuit. FIG.9 shows a generic configuration of a N-state stochastic system, which can be in a ground state S0 and a first state S1, but can further be extended to N-discrete states. Transitions between states are represented by the solid lines. Without loss of generality, it is assumed that there exists at least one sequence of transitions connecting any pair of states. Two arbitrary sites, ^ and ^, are connected by a transition for which we can define the probability current ^^^. The time evolution caused by the transitions between these states is described by a master equation ^(̇t)= ^(^)^(^), where P(^) is the probability vector with components ^^(^), ^ ∈ {1, ... N}, representing the probability for the system to be in state ^ at time ^. W(^)is a stochastic N × N matrix whose elements ^^^(^),(^ ≠ ^)are the transition rates to go from state ^ to state ^ at time ^. To keep the discussion general, these rates are assumed to satisfy the following two (minimal) conditions: (i) in the absence of an external driving, the system evolves towards a unique state of thermal equilibrium, and (ii) for those transitions affected by the driving, the rates depend continuously on the term ^^(^)where ^ is a measure of the strength of the external driving, and ^(^)a periodic function to be specified later. Firstly, the focus will be on the time dependence of the probability currents, given by ^^^(^)= ^^^(^)^^(^)− ^^^(^)^^(^). When considering the long-time limit, that is, the state of the system after all transients have vanished, the former equation and the corresponding currents become periodic in time in this limit. Providing a solution for general time-dependent rates and system size N is computationally complex. However, as will be shown below, analytical expressions can be obtained in the regime of small external driving, i.e., linear response. Given the two minimal conditions, the first-order approximation of the transition matrix follows immediately: ^(^)≈ ^^^+ ^^^^(^). The equilibrium matrix ^^^serves to define the equilibrium state ^^^as the solution of ^^^^^^= 0. Following this, the first-order time-periodic solution of the master equation can be expressed solely in terms of the adiabatic solution ^^^(^), which is defined as the solution to ^(^)^^^(^)= 0,yielding^(^) ≈ ^^^^ (^) −∫^^^τ^^ ^^^^(t − τ). This expression can be further expanded by evaluating the matrix exponential ^^^^^and then performing the matrix product. Since ^^^^is diagonalizable, it has a full set of eigenvalues λ^and eigenvectors^^, where^ ∈ {1, ... N}. Additionally, all eigenvalues are negative except for one, which is zero. The eigenvector associated with the zero eigenvalue is the equilibrium solution (λ^= 0 and ^^= ^^^). Considering the adiabatic solution to be a first-order perturbation of this equilibrium solution, ^^^(t) ≈ ^^^+ ^^^^^^(^), and carrying out the diagonalization, the following expression is obtained: ^( ) ( )^ t ≈ ^^^^ t 〈^^|^〉^^(̇^ − τ). In the former, the inner product is defined Note that〈^^|^^^^〉=〈^^^|^^^^〉= 0 due to normalization. The matrix ^^^is Hermitian with respect to this inner product, and hence the eigenvectors are orthonormal, so〈^^|^^〉= δ^^. Returning to the probability currents equation, and keeping the latter expression in mind, it is noted that the right-hand side can be interpreted as an adiabatic part ^^^(^) and a non-adiabatic part ^^^^(^). Correspondingly, the current can also be split as such, ^^^(^)= ^^^^^(^)+ ^^^^^^(^).The adiabatic part of the current, up to the first order, can be found in ^, = shorthand for taking the previous two terms in its expression and swapping the indices ^ and ^. The non-adiabatic part of the current can be found in At this point, having obtained the general time-dependent first-order expression for the currents, it is advantageous to adopt a complex notation, as is done, for example, in the context of electrical circuits driven by an alternating voltage. This allows the combination of both amplitude and phase into a single complex quantity. For the driving and current, ^^(^)= Re F^^^^and ^^^(^)= Re^^^^^^^^^. The impedance ^ can then be determined by the ratio with coefficients The present impedance equation is the central focus of this embodiment. It is worth noting that while the coefficients ^ are expressed in terms of the eigenvectors of ^^^, they are independent of the choice eigenbasis. The former equations establish a linear relationship between the current and the driving force. This relationship resembles the one found in electrical networks governed by Ohm's law. In electrical systems with alternating voltages, the impedance Z is defined as: ^ = ^^, where ^ is the applied voltage, and ^ is the current flowing through the network. The analogy with the above impedance equation becomes evident in this context. Moreover, a comparison with the impedance addition law for parallel circuits can be made, which states that the total impedance of ^ circuits in ^ ^ ∑ ^ ^ parallel is given by: =^=1^^, where ^^represents the impedance of the ^-th individual component. ^^ The total impedance can then be rewritten as:^, can be interpreted as the impedance of a parallel circuit consisting of ^-components with individual ^ ^ impedances:^^^,^ = ^( )− ^^(^). These individual impedances can be further interpreted as ^^ ^^^ 1 impedance of a device consisting of a resistor ^ ^^,^= ( ) ^ ^ and a capacitor ^^^,^^^= −^^, connected in series. FIG. 10 shows the full equivalent electrical circuit with an impedance that the same form as the above rewritten impedance equation. For notational simplicity, the indices ^^ on the coefficients have been dropped. The circuit includes multiple parallel branches represented by the lines, and their structure is identical to the drawn branches. The equivalent electrical circuit allows the system to be analysed in order to understand its behaviour in terms of impedance and electrical components. As used herein, the terms “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present disclosure. Thus, appearances of the phrases “in some embodiments” in various places throughout this specification are not necessarily all referring to the same embodiments. As used herein, the terms “comprising”, “comprises” and “comprised of” are synonymous with “including”, “includes” or “containing”, “contains”, and are inclusive or open-ended and do not exclude additional, non-recited members, elements or method steps. The terms “comprising”, “comprises” and “comprised of” when referring to recited members, elements or method steps also include embodiments which “consist of” said recited members, elements or method steps. The singular forms “a”, “an”, and “the” include both singular and plural referents unless the context clearly dictates otherwise. As used herein, the terms “connected” or “coupled” reflect a functional relationship between the described objects or devices, that is, the terms indicate the described objects must be connected in a way to perform a designated function which may include a direct or indirect connection in an electrical or nonelectrical (i.e. physical) manner, as appropriate for the context in which the term is used. As used herein, the term “substantially” refers to the complete or nearly complete extent or degree of an action, characteristic, property, state, structure, item, or result. For example, an object that is “substantially” enclosed would mean that the object is either completely enclosed or nearly completely enclosed. The exact allowable degree of deviation from absolute completeness may in some cases depend on the specific context. However, generally speaking the nearness of completion will be so as to have the same overall result as if absolute and total completion were obtained. The use of “substantially” is equally applicable when used in a negative connotation to refer to the complete or near complete lack of an action, characteristic, property, state, structure, item, or result. As used herein, the term “about” is used to provide flexibility to a numerical value or range endpoint by providing that a given value may be “a little above” or “a little below” said value or endpoint, depending on the specific context. Unless otherwise stated, use of the term “about” in accordance with a specific number or numerical range should also be understood to provide support for such numerical terms or range without the term “about”. For example, the recitation of “about 30” should be construed as not only providing support for values a little above and a little below 30, but also for the actual numerical value of 30 as well. The recitation of numerical ranges by endpoints includes all numbers and fractions subsumed within the respective ranges, as well as the recited endpoints. Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequential or chronological order, unless specified. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the disclosure described herein are capable of operation in other sequences than described or illustrated herein. As used herein, the term “improved” with reference to the performance of devices or objects is a measure of a benefit obtained based on a comparison to similar devices or objects in the prior art. Furthermore, it is to be understood that the degree of improved performance may vary between disclosed embodiments and that no equality or consistency in the amount, degree, or realization of improved performance is to be assumed as universally applicable. EXAMPLES To further the understanding of the present invention, a number of exemplary implementations of the present invention will be described below. The provision of these examples is to illustrate the universal applicability of the present invention to stochastic systems of varying complexity. While these embodiments represent possible implementations of the invention, it should be noted that the method is, in principle, not limited solely to these examples. Example 1: Quantification of reaction rates of a two-state model As a proof of concept, a numeric simulation of a two-state model was carried out using realistic values obtained from Stancescu et al. in J. Am. Chem. Soc. 2016, 138, 41, 13465–13468. The simulation mimics an exemplary implementation wherein a DNA probe hybridises to target DNA, as was previously discussed with reference to FIG.5 and FIG.6. The Master Equation was numerically integrated while perturbing the temperature harmonically. The free energy ΔG is determined by enthalpy and entropy differences between the states, which were set to ΔH = 135.7 kcal / mol and ΔS = 0.4334 kcal / mol / K. These values are representative of commonly used 20 base-pair long molecular beacons. The on-rate was chosen to be R = 105 s−1and the simulation was performed at target DNA concentration of c = 1 μM, chosen to optimise experimental conditions. The system could then be perturbed around a reference temperature Tref= 23℃ and perturbation strength δT = 1℃. This determined the temperature dependent off-rate to be ^^ = 0.1727 s−1. The driving frequency ω was varied within a predetermined frequency range of ^ = 0.01 s−1to 10 s−1(of order minute to second periods). Below the results are discussed below discussed for a selected frequency ^ = 0.6 with reference to FIG.11 and FIG.12, and the entire frequency range of ^ = 0.01 s−1to 10 s−1with reference FIG.13 and FIG.14. FIG.11 shows the occupation of the unbound (fluorescent) configuration in function of time for a simulation where the system was perturbed at frequency ^ = 0.6 and a period of about 10 sec. State 1 occupancy (energetically high, dissociated, fluorescent configuration) is shown oscillating around the unperturbed equilibrium value ^ ^^ ^ . FIG.12 shows the response of the experimentally accessible signal 1 − ^^^to the applied temperature perturbation. It can be seen that the rising part of the temperature oscillation corresponds to a decrease in the signal, as more DNA complexes will dissociate and increase the fluorescent signal. Two dashed lines indicate a phase shift between the response of the system and the applied perturbation. This shift, together with the amplitude of the signal, changes in function of driving frequency. FIG.13 shows the frequency dependence of the amplitude response, where numeric results in the range ω = 0.01 s−1to 10 s−1(of order minute to second periods) are compared to the theoretical equation 1 − Further, FIG. 14 displays the frequency dependence of phase shift response for the same range. In FIG. 14 can be seen that there is a steep transition at ω^^≡ R^(1 + A^) = ^^ + ^^, the characteristic timescale of the system. The value of ^^ + ^^ is easily obtained from the frequency ω at which the phase shift reaches It is therefore indicative that a realistic two-state can be analysed on experimentally viable timescales. The curves in FIG. 13 and FIG. 14 determine the two quantities of maximal amplitude, given by δ^ (^^^)^, and the characteristic frequency, given by ^^ + ^^. The function ω^^(^) is linear and has slope ^ and intercept ^^, such that linear regression determines these values. Similarly, measuring ω^^in just two experiments at c1 and c2 fix the values for ^ and ^, denoting ω^^(^^)= λ^and ω^^(^^)= λ^, by the following formulas: ^ = Therefore, measuring ω^^at two experiments at different concentrations allows the determination of rates ^ and ^^. Once on- and off-rate ^ and ^^ are known, the maximal amplitude of oscillation, given by the above-referenced formula, also fixes the value of a. The relation between A and a finally allows the extraction of thermodynamic parameters ΔH and ΔS, given by ΔH = ^^^^^^ and ^^ ^^ ^ ^ ΔS = ^^ln(^)+ ^^^ ^, which can be derived from the following expressions: ^ = ^^^^^^^and ^^ ^ = ^^^^^. It may be noted that for in which the entropy of the two states remains the same, the free energy difference ΔG is just equal to the energy difference between the states ΔE. In such a system a single experiment suffices to uniquely determine ΔE, as the maximal amplitude is dependent on ^(ΔE) and ^(ΔE). Present Example 1 therefore demonstrates that a frequency analysis at multiple concentrations allows the extraction of forward and backward reaction rates. Returning back the DNA example, these reaction rates are explicitly related to enthalpy and entropy difference ΔH and ΔS. Therefore, a way of determining reaction rates and measuring DNA thermodynamic parameters is realised. As a consequence, an aspect of the present invention can be implemented as a measuring system for kinetic and thermodynamic properties of a molecular system, such as a DNA probe / target complex. Example 2: Distinction between DNA point mutations As a further example, the present invention can be implemented as a detection device, for instance, to identify (small) alterations in molecules. One specific example is the detection of point mutations in DNA sequences, which play a major role in diagnostics. Point mutations can be inherited single nucleotide variants that provide important information about predispositions for certain diseases. They can also be somatic mutations, occurring during tumour development, and offer crucial insights into therapy selection and disease progression. As an exemplary implementation, the detection of mutated DNA from wild-type DNA can be considered. The setup is similar to Example 1, where probes are labelled with a fluorophore and a target DNA is labelled with a quenching molecule, as shown in FIG.6. The unbound states are fluorescent, while bound states are not. The target DNA consists of a perfectly matching wild-type sequence in one case and the mutated DNA with a single nucleotide mismatch in the other case. As a proof of concept, simulations were carried out on the epidermal growth factor receptor (EGFR) and its mutation T790M, an important biomarker for decisions on the follow-up treatment of lung cancer. FIG 15 shows simulations of the fluorescent amplitude response against the driving frequency of pure wild-type and pure mutant. Further, FIG.16 displays the phase shift response for the same. The theory is indicated by the full line. A frequency analysis allows distinguishing between the two. Two features clearly distinguish the wild type from the mutant: the characteristic frequency (indicated by the dashed verticals) and the maximum amplitude. While the latter depends on the concentration used (and as a consequence is more prone to experimental fluctuations), the characteristic frequency is completely determined by the kinetic properties of the molecular process and can serve as a robust detection feature. Present Example 2 therefore demonstrates that the presence of (small) molecular changes in DNA sequences can be effectively detected. As a consequence, an aspect of the present invention can be implemented as a detection system for identifying (small) molecular changes in a varied molecular system, such as identifying point mutations in a DNA sequence. Example 3: Detection of DNA nucleotide mismatches in a mixture As a further example, the present invention can be implemented as a detection device, particularly to identify nucleotide mismatches in a mixture. In clinical samples with somatic mutations, it is highly important to detect the presence of mutant DNA in such mixtures. The setup is similar to Example 1, and building upon Example 2, it is demonstrated that the frequency response of pure wild-type and pure mutant samples differs. This finding can now be extended to a mixture, where a (possibly small) amount of mutant DNA is present alongside a majority of wild-type DNA. FIG.17 shows simulations of the amplitude response against the driving frequency for the three cases: wild-type only, mutant only, and the mixture. Further, FIG.18 displays the phase shift response for the same. In the mixture, a distinct deviation from the response of a pure sample is visible at the characteristic frequency of the mutant transition. Present Example 3 therefore demonstrates that the presence of mutant DNA in the mixture can be effectively detected. As a consequence, an aspect of the present invention can be implemented as a detection system for identifying different molecules in a mixed molecular system, such as a identifying the presence of mutant DNA in a mixture. Example 4: Detection of hidden states or intermediate reaction steps Many chemical reactions consist of a sequence of intermediate reaction steps. The detection of intermediate states is key to determining accurate reaction profiles in synthetic chemistry, and it plays an important role in understanding various chemical processes, such as the formation of free radicals. Additionally, detection of intermediate states can aid in the development of better performing biosensors. The present invention can be used to identify whether a process is the result of a multi-step reaction. To illustrate this, a simple example is presented where a reaction occurs via one intermediate step, resulting in a three-state model as depicted in FIG.3. However, only the high- energy state, E2, can be detected by a fluorescent signal, leaving the intermediate state essentially hidden from (direct) detection. In equilibrium, the system appears indistinguishable from a two- level system with effective forward and backward rates. Nevertheless, the present invention, through its spectral analysis, may allow the detection of the hidden state. FIG.19 shows simulations of the amplitude response against the driving frequency of the three- state system (with a 'hidden' intermediate state) compared to a simple two-state system. Further, FIG. 20 shows the phase shift response for the same. The two-state system exhibits a monotone ^ phase shift, varying between 0 and ^. In contrast, the system with a hidden state deviates from this behaviour. FIG 14 shows simulations of the amplitude response against the driving frequency of a three-state system (with a ‘hidden’ intermediate state) compared to a simple two-state system, and FIG.15 further shows the phase shift response. The two-state system shows a monotone phase shift ^ varying between 0 and ^. The system with a hidden state deviates from this behaviour. Focusing further on the phase plot in FIG.15, the phase shift for a two-state system was demonstrated to have frequency dependence according to ^ = arctan Consequently, the ^ phase shift ranges from 0 to ^ and is monotone. The two-state simulations of Example 1 clearly showed this behaviour in FIG 9. However, this is in contrast to what can be observed for the three- state system. As a result, it is possible to distinguish a two-state system from a three-state system, and consequently, detect 'hidden' states by carrying out a frequency analysis. Present Example 3 therefore demonstrates that hidden or intermediate states in a multi-step chemical reaction can be effectively detected. As a consequence, an aspect of the present invention can be implemented as a detection system for identifying hidden states in a stochastic system that can be in three or more states, such as a identifying the presence of intermediate states in a multi- step chemical reaction.

Claims

CLAIMS 1. Method (100) for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, comprising the steps of: - providing the stochastic system; - generating an equivalent electric circuit that represents a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit comprises a plurality of electrical components; wherein the equivalent electric circuit has an impedance ^ related to the transition rates of the stochastic system; - applying a periodic perturbation ^ to the stochastic system, wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; - measuring an impedance response of the stochastic system to the periodic perturbation in a stationary time-periodic state, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the electrical component of the equivalent electric circuit to match the impedance ^ with the frequency-dependence of the amplitude and phase shift of the measured impedance response; and, - quantifying the transition rates of the stochastic system based on the equivalent electric circuit.

2. The method according to claim 1, wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system.

3. The method according to claim 2, wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system.

4. The method according to any one of the preceding claims, wherein the stochastic system is a two-state stochastic system, comprising a single component that can be in a ground state S0 and a first state S1, wherein the transition rates are quantified by - determining the ratio A of the ground state S0 and the first state S1 rate; - determining a characteristic frequency ^, wherein the characteristic frequency is determined by plotting the amplitude and the phase shift as functions of the drivingfrequency ω, and selecting the frequency for which the phase shift substantially equals 45 degrees; - calculating the transition rate R from the first state S1 to the ground state S0 using R =- calculating the transition rate from the ground state S0 to the first state S1 based as R . ^. 5 The method according to any one of the preceding claims, wherein the stochastic system is a two-state stochastic system, comprising a plurality of components that can interact with each other to form a ground state S0 and a first state S1; wherein the impedance response of the stochastic system to the periodic perturbation is measured for at least two different concentrations of at least one of the components of the stochastic system, ^^and ^^; wherein the transition rates are quantified by - determining a characteristic frequency for each of the at least two different concentrations,- calculating the transition rate R from the first state S1 to the ground state S0 using ^^(^^^^^) R = ^^^^^^^^^, - calculating the ratio A of the ground state S0 and the first state S1 rate using- calculating the transition rate from the ground state S0 to the first state S1 as R . ^.

6. The method according to claim 5, wherein the components include at least one probe component and at least one target component, wherein the probe is configured to interact with the target component to form one of the states of the stochastic system; and wherein the perturbation is measured for at least two different concentrations of the target component; preferably wherein the probe component comprises a probe molecule and the target component comprises a target molecule, wherein the probe molecule is configured to interact with the target molecule to form a complex.

7. The method according to any one of claims 5 or 6, wherein the characteristic frequency is determined by plotting the amplitude and the phase shift as functions of the driving frequency ω, and identifying a crossover point in the phase shift plot; preferably whereinthe crossover point is characterized by a sudden change or discontinuity in the slope of the phase response curve.

8. The method according to any one of the preceding claims, wherein the periodic perturbation comprises one or more types of perturbations selected from the group consisting of harmonic perturbations, step perturbations, pulse perturbations, periodic perturbations with non-harmonic waveforms; preferably step perturbations.

9. The method according to any one of the preceding claims, wherein the periodic perturbation is applied by varying at least one parameter selected from the group consisting of temperature, pressure, concentration, electromagnetic radiation preferably comprising one or more wavelengths within the visible, ultraviolet, and / or infrared spectrum, and / or a combination thereof, to induce the impedance response in the stochastic system; preferably temperature.

10. The method according to any one of the preceding claims, wherein the periodic perturbation is applied in discrete steps within the predetermined frequency range, with each step maintaining a constant driving frequency for a predefined duration before transitioning to the next driving frequency; preferably wherein the constant driving frequency is maintained until the stochastic system reaches a stationary time periodic state.

11. The method according to any one of the preceding claims, wherein the stochastic system comprises deoxyribonucleic acid (DNA), and wherein the method further comprises identifying mutations in a sequence of the DNA and / or distinguishing between different types of DNA based, on the quantified transition rates of the stochastic system; and / or, wherein the stochastic system comprises a chemical reaction comprising a sequence of one or more intermediate reaction steps, and wherein the method further comprises identifying the presence of the one or more intermediate reaction steps based on the quantified transition rates of the stochastic system.

12. Method for quantifying the change of enthalpy ΔH and / or the change in entropy ΔS of a stochastic system that can be in a plurality of states, comprising the steps of: - performing the method of any one of the preceding claims to quantify the transition rates of the stochastic system; wherein the stochastic system is provided at a reference temperature T^^^; wherein the applied periodical perturbation comprises a thermal perturbation based on T(^)= T^^^+ ^T(^), wherein ^^ is the perturbation strength around the reference temperature; and,- quantifying the change of enthalpy ΔH and / or the change in entropy ΔS for the stochastic system based on the quantified transition rates of the stochastic system and the reference temperature.

13. The method according to claim 12, wherein the system is a two-state stochastic system, comprising a single component that can be in a ground state S0 and a first, higher energy state S1, or a plurality of components that can interact with each other to form the ground state S0 and the first state S1; wherein the impedance response of the stochastic system is measured for at least two different concentrations of the stochastic system, ^^and ^^; wherein the transition rates are quantified by - determining a characteristic frequency for each of the at least two different concentrations, ^^and ^^; - calculating the transition rate R from the first state S1 to the ground state S0 based on ^ (^ ^^ ) R =^ ^ ^^^^^^^^^^; - calculating the ratio ^ of the ground state S0 and the first state S1 rate based on- calculating the transition rate from the ground state S0 to the first state S1 based on R . A. wherein the change of enthalpy ΔH and / or the change in entropy ΔS is quantified by - determining a maximum amplitude û^^^based on the plurality of amplitudes û in the measured impedance responses; - calculating the ratio ^ based - calculating ΔH based on ΔH - calculating ΔS based on ΔSwherein ^^is the Boltzmann factor.

14. Computer-implemented method for quantifying a plurality of transition rates of a stochastic system that can be in a plurality of states, comprising the steps of: - generating an equivalent electric circuit that represents a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor andcapacitor equal to the impedance ^ associated with one state transition of the stochastic system; - receiving sensor data indicative of an impedance response of the stochastic system to a periodic perturbation; wherein the periodic perturbation has a driving frequency ω that is varied within a predetermined frequency range; wherein the impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - mapping the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; - quantifying the transition rates of the stochastic system based on the equivalent electric circuit; and, - optionally, outputting the quantified transition rates of the stochastic system.

15. System for quantifying a plurality of transitions rates of a stochastic system that can be in a plurality of states, comprising: - a perturbator means configured for applying a periodic perturbation to the stochastic system, wherein the periodic perturbation has a driving frequency ω that can be varied over a predetermined frequency range; - a sensor means configured for measuring an impedance response of the stochastic system to the periodic perturbation, wherein the measured impedance response includes at least an amplitude and a phase shift as functions of the driving frequency ω; - optionally, a receiver means adapted for receiving the stochastic system, wherein the perturbator means and the sensor means are disposed on the receiver means; and, - a processing unit, optionally comprising an output means, wherein the processing unit is communicatively connected to the perturbator means and sensor means, and configured to: - generate an equivalent electric circuit that represents a probability current associated with the state transitions in the stochastic system; wherein the equivalent electric circuit consists of a plurality of parallel branches, with the number of parallel branches equal to the number of states in the stochastic system; wherein each parallel branch includes a resistor ^ and a capacitor ^ connected in series, with the values of the resistor and capacitor equal to the impedance ^ associated with one state transition of the stochastic system;- map the values of the resistor and the capacitor of each parallel branch of the equivalent electric circuit such that the impedance of the equivalent electric circuit matches the amplitude and phase shift of the measured impedance response; - quantify the transition rates of the stochastic system based on the equivalent electric circuit; and, - optionally, output the quantified transition rates of the stochastic system.

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