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A Unified Approach for Investigating Chemosensor Properties – Dynamic Characteristics

Christian G. Frankær†*

and Thomas Just Sørensen†*

Contents

Contents ............................................................................................................................................................ 1

Dynamic characteristics of commercial pH-electrodes....................................................................................................... 2

Definitions of terms ............................................................................................................................................... 3

Deriving linear time invariant model ........................................................................................................................... 7

Relation between conversion fraction, time constant, and differential quotient used for curve fitting .............................................. 8

Reporting dynamic properties of pH electrode ............................................................................................................... 9

References ........................................................................................................................................................ 12

Electronic Supplementary Material (ESI) for Analyst.This journal is © The Royal Society of Chemistry 2019

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Dynamic characteristics of commercial pH-electrodes

Table S1. Response characteristics of commercial pH electrodes.

Manufacturer Method Quantity reported Value

Jenway1

a) Electrode preequilibrated in pH 7 buffer at 20°C, 200 rpm,

then rinsed and measured in pH 4 buffer at 20°C, 200 rpm.

b) Electrode preequilibrated in pH 4 buffer at 20°C, 200 rpm,

then rinsed and measured in pH 4 buffer at 60°C, 200 rpm.

Stabilized reading, tstab, for 10

seconds (stabilization criterion

not reported)

a) tstab < 20-30 s

b) tstab < 30-60 s

Agilent2

a) Electrode preequilibrated in buffer pH 4.01 at 24.8°C, then

rinsed and measured in pH 10.01 buffer, then rinsed and

measured in pH 7.00 buffer. Stirring applied, but no

specifications reported.

b) Electrode immersed in pure water (low conductivity).

Container is sealed to shield the samples from carbondioxide

uptake and other contamination. No stirring.

Stabilized reading, t(ΔE/Δt),

defined as no change by more

than 0.002 pH units during 7

seconds

a) t(ΔE/Δt) < 29-33 s

b) t(ΔE/Δt) ~ 300 s

ThermoFisher

Scientic3, 4

a) AccupHast-electrode: Preequilibrated in buffer pH 7.00 then

rinsed and measured in buffer pH 4.01.

b) ROSS Ultra, ROSS Ultra Triode and ROSS pH Electrodes: pH

change in buffers between pH 6.86 and pH 4.01 at 25°C

c) ROSS Ultra, ROSS Ultra Triode and ROSS pH Electrodes:

Temperature change in buffer pH 6.86 between 25°C to 75°C.

a) t95

b) Stabilized reading, t(ΔE/Δt),

defined as no change by more

than 0.005 pH units

c) Stabilized reading, t(ΔE/Δt),

defined as no change by more

than 0.01 pH units

a) t95 < 10 s

b) t(ΔE/Δt) < 30 s

c) t(ΔE/Δt) < 30 s

Walchem5 Change between two buffers used in two-point calibration.

Agitation applied, but not specification reported.

Stabilized reading, tstab,

(stabilization criterion not

reported)

tstab < 10-15 s for new

electrodes

Hamilton6 pH change between pH 4 and 7 or 7 and 10. No specifications

on agitation reported.

Stabilized reading, t(ΔE/Δt),

defined as no change by more

than 0.01 pH units

t(ΔE/Δt) ~ 30 s

Mettler Toledo7 No description of method in details. However, emphasis on

defining measuring conditions, particularly temperature and

homogeneity of the sample solution.

Stabilized reading, t(ΔE/Δt), with

a tolerance of 0.01 pH units

within the range of standard

buffers (pH 4 to 9).

t(ΔE/Δt) < 5 s for new

electrodes

Yokogawa8-11

a) General: pH step change between pH 1.68 and 7.

b) FU20/FU24 electrodes: pH step change from pH 7 to 4.

c) pH20 electrodes: pH step change from pH 7 to 4.

d) SC25V electrode: pH step change from pH 7 to 4.

a) τ = t63 upon step change

b-d) t90 and stabilized reading

t(ΔE/Δt) defined as no change by

more than 0.02 pH units during

10 s

a) τ < 5 s

b) t90 < 15 s, t(ΔE/Δt) < 120 s

c) t90 < 10 s, t(ΔE/Δt) < 60 s

d) t90 < 5 s, t(ΔE/Δt) < 120 s

Honeywell12 Durafet III electrodes: No description of method Stable reading at pH 7, tstab,

(stabilization criterion not

reported )

2-10 s

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Definitions of terms

Table S2: Definition of terms

Term Definition

Activity step change The input change concerning activity change by a step function.

Activity step method Method to determine dynamic sensor characteristics in which a step change of the input variable (the

activity of the target analyte) is introduced.13

Analyte The analyte indicate the chemical entity whose activity is reported by the chemosensor.14 This is

typically the chemical species of interest in an analytical procedure, also mentioned as the target

species, e.g. for a pH sensor, the analyte is protons (hydrons).

Bias See systematic error

Calibration curve Fitting of a model to the response curve yields the calibration function. A calibration curve is a plot of

the calibration function and describes the sensor output signal versus the logarithm of the analyte

concentration/activity.15 A calibration curve of a pH sensor correlates pH with the output generated

from the sensor, and provides information about the limit of detection, operational range, sensitivity,

and linearity. Comparison between calibration curve and response curve provides information about

precision.

Chemical sensor Identical to chemosensor

Chemosensor An analytical small-sized device by which chemical information, ranging from concentration of a

specific compound or ion (commonly termed the analyte or measurand) to a total composition

analysis, can be transformed into an analytically useful signal and delivered in real time.16-18

Conversion fraction, α It is defined: 𝛼 = (𝑆 − 𝑆0)/(𝑆∞ − 𝑆0) and is analogue to the normalized response signal. Included in

the tα definition of response time.

Dead time Dead time, tdead, is the length of time from the instant when the activity change is introduced to the

solution by the operator, and the instant at which the activity change occurs at the solution-

membrane interface (t = 0). This is most prominent for the injection-based methods.

Detection limit Minimal and maximal detectable analyte concentration by the sensor. The detection limits can be

determined from the calibration curve, and depends on the error, e. Explicit definitions can be found

in reference15.

Determinand Another term for measurand.

Dipping method One of the two methods recommended by IUPAC to determine the response time of a sensor, namely

the static response time. Using this method, the electrode/optode is instantaneously immersed into a

sample solution with known analyte activity after it has been pre-conditioned in another solution of

known composition The electrode/optode may be gently wiped with a tissue paper to minimize

contamination. At the instant where it is brought in contact with the sample solution (t = 0) recording

of time-response curve data is started. Moderate agitation is applied throughout the process.13, 19

Drift Slow non-random change with time in the output signal from the chemosensor operating in a solution

of constant composition and temperature. Drift is always characterized by a linear curve fitting on the

data set collected in a given period of time in a solution of constant composition and temperature,

and defined as the slope of the line fitted to the (time vs. signal)-plot.15

Dynamic characteristics Characteristics describing the transient properties of a sensing system, i.e. how the system is

responding during a change in measurand. Dynamic characteristics exist because of energy-storing

elements in the sensor.20 These elements could be of electronic, mechanical or thermal character,

and for chemosensors, typically related to mass transfer. Dynamic characteristics are determined

from the time-response curve, with response time and stabilization time being the most important

descriptive parameters.

Dynamic response time Response time measured by the injection method.

Electrode The part of an electrochemical chemosensor that is in contact with the sample solution. Analogue to

optode.

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Error, e The difference between the observed value and the true value. The error has two components:

random error and systematic error.14

Experimental conditions The conditions present in the sample solution surrounding the electrode or optode during recording

of a time-response curve. These conditions are provided by the experimental setup, and they cover

compositional, hydrodynamic and thermal conditions.

Experimental setup Setup used for dynamic characterization of a sensor, i.e. recording of a time-response curve. The

experimental setup consists of the sample solution, its container and the equipment used for agitating

the solution. Flow-cells are also defined as experimental setup.

Injection method One of the two methods recommended by IUPAC to determine the response time of a sensor, namely

the dynamic response time. Using this method, the electrode/optode is maintained within the

solution in which it has been pre-conditioned under good agitation. The recording of response data is

stated at t < 0, under the assumption of steady state. At t = 0 a small aliquot of a concentrated

solution of the target analyte is rapidly injected and the measured variable is changed.13, 21

Input change, x The change in a physical or chemical property in the environment on which the sensor is tested. For

chemosensors the input change is typically activity change of the target analyte.

Input function, x(t) Input change as function of time, t.

Input variable Input change as a function of time described by the input function x(t)

Lag time The time elapsed from the instant at which the activity change occurs at the solution-membrane

interface (t = 0) to the instant, when a significant change in signal output y(t) is observed from the

sensor, is described by lag time, tlag. tlag is defined as the intersection between a line tangential to y(t)

with the highest possible slope (either positive or negative for respectively analyte activity increase

and decrease) and the baseline.

Limit of detection See detection limit

Limiting mean, μ The value that is approached as the number of observations approaches infinity.14

Linearity The closeness of the calibration curve to a specified straight line.20 The linear range of a sensor is

defined as the region of the calibration curve that can be well approximated by a linear curve fit.

Matrix material The supporting material without responsive molecules, e.g. ORMOSIL-networks

Measurand Another more encompassing term for analyte14, but includes non-chemical entities such as

temperature, pressure, etc.

Measured variable See input variable

Noise See random errors

Operational range The operational range of a sensor is defined as the region of the calibration curve between the lower

and upper detection limit.

Optode The part of an optical chemosensor that is in contact with the sample solution. The optode contains

responsive molecules immobilized in a matrix material, which is deposited on a robust substrate.

Analogue to electrode.

Output change, y The change in signal output from the sensor that is tested. For electrode based chemosensors the

output change is typically change in potential created by the the input change. For optode based

chemosensors the output change is for example change in optical property generated by the

transducing part of the sensor upon an input change.

Output function, y(t) Output change as function of time, t.

Output variable Output change as a function of time described by the output function y(t)

Random errors Random errors or noise are the difference between an observed value and the limiting mean.14

Random errors constitute the inherently unpredictable fluctuations in the readings. They are

unbiased and normally distributed with the standard deviation σ.

Reading time The time interval that, when employed in a homogeneous series of measurements, including

calibration, combines acceptable precision with time savings.22 The reading time is always equal to or

longer than the stabilization time with a given stabilization criterion that defines the acceptable

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precision.

Receptor A receptor is the first sensor element being the part of the sensor in which the chemical information

(analyte concentration/activity) is transformed into a form of energy which may be measured by the

transducer.16

Recognition system Another term covering the receptor element.

Reproducibility The systematic error observed between signal measurements performed in the same sample solution,

but after the sensor has been exposed to different conditions between each measurement.20

Response curve A plot of sensor output signal versus the logarithm of the analyte concentration/activity. A response

curve of a pH sensor correlates pH with the output generated from the sensor. Note that the

response curve should not be confused with the time-response curve.

Response signal, S(t) Modelled signal output obtained during curve fitting to time-response curve data.

Response time A parameter used to describe the transient function. Three definitions are used:

t*: “The length of time which elapses between the instant at which an ion-selective electrode

and a reference electrode are brought into contact with a sample solution and the first

instant at which the cell becomes equal to its steady-state value within 1 mV”.23

tα (e.g. t90): “The length of time which elapses between the instant at which an ion-selective

electrode and a reference electrode are brought into contact with a sample solution and the

first instant when the potential of the cell reached 90% of the final value”.24

t(ΔE/Δt): “The time which elapses between the instant when an ion-selective electrode and a

reference electrode are brought into contact with a sample solution and the first instant at

which the emf/time slope (ΔE/Δt) becomes equal to a limiting value selected on basis of the

experimental conditions and/or requirements concerning the accuracy (e.g. 0.6 mV/min)”.15,

25

Responsive molecules

Compounds susceptible to respond upon change in chemical conditions, i.e. concentration/activity of

a target analyte. For instance, responsive molecules are the active dye components used in optical

chemosensors

Sample solution The solution in which the electrode or optode is immersed during recording of time-response curves

and calibration curves.

Selectivity The ability of the sensor to distinguish the target analyte in presence of other interferences.

Sensitivity The ratio between the incremental change of the output signal, Δy, and the corresponding

incremental change of the input, Δx. In other words, the tangential slope of the calibration curve. A

sensor is said to have high sensitivity in the regions of the calibration curve where this slope is large

(either negative or positive).

Sensor material The matrix material with responsive molecules

Sensor spot Key component of a sensor being in contact with the sample solution, defined as a device at which

the recognition and signal transduction takes place.

Sensor setup The optode/electrode, hardware and software.

Sensor signal The signal output form the sensor

Signal-to-noise ratio The noise relative to the signal. For time-response curves the signal is the change in signal output, y,

upon an input change, x.

Signal output See output change

Signal processor The signal processor is the third and final element of a sensor capable of amplifying and processing

the signal from the transducer into meaningful readouts.

Signal variable See output variable.

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Stability criterion (ΔE/Δt) The limiting value of the (ΔE/Δt) slope (or (ΔS/Δt)-slope) that has been selected. Given in signal units

per time units.

Stabilization time, tstab A parameter used to describe the transient function. With a given stability criterion (ΔE/Δt) the

stabilization time is the same as response time using the t(ΔE/Δt)-definition. When no stability

criterion is reported, stabilization time is denoted tstab (however this parameter is not of much use).

Standard deviation σ The standard deviation of a measurement reflects the root mean square random deviation of the

observations about the limiting mean.14

Static characteristics Characteristics describing a sensor’s properties at stable or steady-state conditions where no change

is occurring. Static characteristics include limit of detection, operational range, sensitivity, linearity,

precision, which are determined from a calibration curve. It also include non-ideal characteristics such

as selectivity, drift and noise.

Static response time Response time measured by the dipping method

Steady-state The system is in a steady-state if the variables defining the behaviour of the system is unchanging

with time. For instance, a steady-state sensor output corresponds to an output from system that is

fully equilibrated.

Systematic errors Systematic errors or bias is the difference between the limiting mean and the true value. Systematic

errors are predictable and caused by imperfections in the calibration, the observation or interference

from the surroundings.

Time-response curve The transient signal output from sensor recorded continuously during an activity step change plotted

as a function of time.

Transducer A transducer is the second sensor element being the part of the sensor capable of transforming the

energy carrying the chemical information (analyte concentration/activity) about the sample into a

useful analytical signal.16

Transfer function The ratio between the input and the output.

Time resolution The frequency by which data points are sampled. The highest possible time resolution is defined by

the sensor setup.

| S7

Deriving linear time invariant model

The ratio between the input and output is called the transfer function.26

In LTI systems, the output signal y(t) from a sensor and

its relationship to the input signal x(t) can be described by the following differential equation:

𝑎𝑛d𝑛𝑦(𝑡)

d𝑡𝑛+ 𝑎𝑛−1

d𝑛−1𝑦(𝑡)

d𝑡𝑛−1+ ⋯ + 𝑎1

d𝑦(𝑡)

d𝑡+ 𝑎0𝑦(𝑡) = 𝑏𝑚

d𝑚−1𝑥(𝑡)

d𝑡𝑚−1+ 𝑏𝑚−1

d𝑚−2𝑥(𝑡)

d𝑡𝑚−2+ ⋯ + 𝑏2

d𝑥(𝑡)

d𝑡+ 𝑏1𝑥(𝑡) + 𝑏0 (S1)

For first order systems the output described by the left-hand side reduces to include only the first derivative terms, and for step

change input signals, the right hand side reduces to b1. And then the resulting differential equation becomes:

𝑎1d𝑦(𝑡)

d𝑡+ 𝑎0𝑦(𝑡) = 𝑏1 (S2)

That rearranged becomes:

𝑎1

𝑎0

d𝑦(𝑡)

d𝑡+ 𝑦(𝑡) =

𝑏1

𝑎0(S3)

When defining the parameters τ = a1/a0, and K = b1/a0 results in the following first-order differential equation:

𝜏d𝑦(𝑡)

d𝑡+ 𝑦(𝑡) = 𝐾 (S4)

Here, the parameter τ is the time constant characteristic for the sensor, where τ is defined as the time it takes the output

variable to reach ~ 63 % of the steady state value, calculated from the time at which the input variable is changed, see Figure 7.

Thus = t63 stated as t for = 63%:

1 − exp (−1) = 0.6321 (S5)

For comparison with other models, we denote the time constant for the LTI model as kLTI:

𝑘LTI =1

𝜏 (S6)

The solution of the first order LTI model corresponds to the kinetics of a first-order reaction, thus applying the LTI model results

in expressions resembling those of first-order kinetics. Solving this first-order differential equation results in:

𝑦(𝑡) = 𝐾 + 𝐴 exp(−kLTI 𝑡) (S7)

Setting K = S∞ and A = (S0 – S∞), this results in a rewritten equation where the signal changes from S0 to S∞, with subscripts 0 and

∞ denoting initial and final steady-state signal values:

𝑆(𝑡) = 𝑆∞ − (𝑆∞ − 𝑆0) exp(−kLTI 𝑡) (1)

Time-response curves for activity step changes that lead to increased signal outputs are normalized so that yNorm

(0) = S0 = 0 and

yNorm

(∞) = S∞ = 1. And response curves recorded for activity steps that lead to decreased output signals are similarly normalized

with yNorm

(0) = S0 = 1 and yNorm

(∞) = S∞ = 0. In the LTI model, response signal S(t) is independent of the activity step and predicts

symmetrical response curves for S(t) with regards to the direction of the activity step, as seen in Figure 7.

The empirical LTI model can be related to empirical data using S. For instance: for sensors that have a response that varies

linearly with the logarithm of the activity with a slope s, S0 can be expressed using the equilibrium signal S∞ and the analyte

activities at t = 0 and at steady state t = ∞. Thus, S0 can be expressed as a function of S∞, s, a0 and a∞ as:

𝑆0 = 𝑆∞ + 𝑠 log(𝑎0/𝑎∞) (S8)

Equation (1a) can then be re-written as:

𝑆(𝑡) = 𝑆∞ + 𝑠 log (𝑎0

𝑎∞) exp(−𝑘𝐿𝑇𝐼 𝑡) (3)

| S8

Relation between conversion fraction, time constant, and differential quotient used for curve fitting

The time constant at different conversion fractions are given in Table S3 for all five models presented for 10-fold increased and decreased activity steps. Similarly, differential quotient at different conversion fractions is given in Table S4. These values are helpful when distinguishing slope from drift.

Table S3: Relation between conversion fraction α and time constant ki for empirical and physical models with 10-fold activity increase, a0/a∞ = 0.1, and 10-fold decrease, a0/a∞ =

10.

Number of 1/k (i.e. τ for the LTI model)

Conversion fraction, α

(i)

First-order LTI

model

(ii)

Hyperbolic

model

(iii)

Boundary layer

diffusion controlled

(BL)*

(iv)

Bulk electrode

membrane controlled

(Mem)*

(v)

Bulk optode diffusion

controlled (Opt)

Eqn. (2) Eqn. (1/3) Eqn. (4a/b) Eqn. (5) Eqn. (7) Eqn. (9)

1/kLTI = τ 1/khyp 1/kBL 1/kMem 1/kOpt

t½ 0.50 0.6931 1 0.2748/1.4261 0.1000/10 0.4853

τ 0.6321 1 1.7181 0.4544/1.9099 0.3309/33.09 0.7903

t90 0.90 2.3026 9 1.4761/3.5484 11.40/1.140×103 2.0925

t95 0.95 2.9957 19 2.1134/4.3008 52.94/5.294×103 2.7858

t99 0.99 4.6052 99 3.6773/5.9568 1.485×103/1.485×105 4.3953

0.995 5.2983 199 4.3647/6.6557 6.025×103/6.025×105 5.0882

0.999 6.9078 999 5.9695/8.2698 1.524×105/1.524×107 6.6977

0.9999 9.2103 9999 8.2711/10.5734 1.527×107/1.527×109 9.0003

*For models providing asymmetrical response with regards to the direction of the activity step, the two values given are respectively for 10-fold activity increase, and

10-fold decrease.

Table S4: Differential quotient dS/dt (tangential slope) at different conversion fraction α for empirical and physical models with 10-fold activity increase, a0/a∞ = 0.1, and 10-fold

decrease, a0/a∞ = 10.

dS/dt

Conversion fraction, α

(i) First-order LTI model

(ii) Hyperbolic model

(iii) Boundary layer

diffusion controlled (BL)*

(iv) Bulk electrode

membrane controlled (Mem)*

(v) Bulk optode diffusion

controlled (Opt)

Eqn. (2) Eqn. (1/3) Eqn. (4a/b) Eqn. (5) Eqn. (7) Eqn. (9)

t½ 0.50 0.5/–0.5 0.25/–0.25 0.9390/–0.2970 1.13/–0.011 0.5091/–0.5091

τ 0.6321 0.3679/–0.3679 0.1353/–0.1353 0.5788/–0.2481 0.32/–3.2×10–3 0.3685/–0.3685

t90 0.90 0.1/–0.1 0.01/–0.01 0.1124/–0.0893 3.8×10–3/–3.8×10–5 0.1/–0.1

t95 0.95 0.05/–0.05 2.5×10–3/–2.5×10–3 0.0530/–0.0472 4.4×10–4/–4.4×10–6 0.05/–0.05

t99 0.99 0.01/–0.01 1.0×10–4/–1.0×10–4 0.0101/–0.0099 3.3×10–6/–1.0×10–8 0.01/–0.01

0.995 0.005/–0.005 2.5×10–5/–2.5×10–5 0.0050/–0.0050 - 0.005/–0.005

0.999 0.001/–0.001 1.0×10–6/–1.0×10–6 0.0010/–0.0010 - 0.001/–0.001

0.9999 0.0001/–0.0001 1.0×10–8/–1.0×10–8 0.0001/–0.0001 - 0.0001/–0.0001

* For models providing asymmetrical response with regards to the direction of the activity step, the two values given are respectively for 10-fold activity increase, and

10-fold decrease.

| S9

Reporting dynamic properties of pH electrode

The following description of the experimental procedure used to characterize the dynamic properties of a pH electrode is an example of how to report the experimental conditions and how to describe the setup used for determination of response time. Here, an activity step change of 10 (pH changed from 8.0 to 7.0) is applied. Both initial and final H+ activities are covered by the operational range of the pH electrode, in which the electrode is linearly responding to changes in pH. The sample solution used was buffered by 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES) having a pKa of 7.55. HEPES has a good buffer capacity in the range in which the activity step is positioned.

Experimental procedure

The injection method was used to determine the dynamic response time of a pH electrode. A sample solution containing 0.020 M 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES) was prepared by dissolving HEPES in deionized water. pH was adjusted to 8.0 with a 1 M NaOH before the solution was brought to its final volume. A bar magnet (length 20 mm, width 6 mm) was placed in a 100 ml beaker (diameter 50 mm, height 80 mm). Then 50 ml sample solution was transferred to the beaker creating a liquid column with a height of 35 mm. The beaker was placed on a magnetic stirrer, and agitation at 400 rpm was applied. The pH-electrode (Mettler-Toledo, InLab Micro Pro) had a diameter of 5 mm, and a surface area of approx. 40 mm2. The pH-electrode was calibrated using a series of four standard technical buffer solutions at pH 2.00, 4.01, 7.00, and 10.00 (Mettler-Toledo, InLab solutions). The pH-electrode was immersed into the beaker 10 mm from the vortex centre with an angle of 90° with respect to the liquid surface and positioned with the tip 20 mm beneath the surface. The electrode was allowed to pre-equilibrate for 10 min. The temperature was recorded simultaneously with the electric potential using a pH meter (Mettler-Toledo Seven Compact). The temperature was constant within 0.1°C throughout the experiment (24.2±0.1°C). The electric potential was read out every second (tres = 1 s) during a period of 1000 s. After 4.5 min (tmin = –270 s), 0.50 ml 1 M HCl was injected 10 mm from the vortex centre using a 100-1000 ml mechanical pipette. The experiment was stopped after 10 min (tmax = 609 s).

The data (Data28062018.txt) is included in the supporting information.

Data fitting and explanatory comments to the algorithm

The algorithm (ResponseCurveFit.m) included as supplementary information is an example of how to extract the key dynamic characteristic, i.e. the response time tα, from a response curve using the procedure outlined in the flow chart in the main text (Figure 11). The first input DataFile is a (t,y)-file containing the response curve data in two columns. Units and axis labels to the plot are defined. Note that the frequency, by which the data points are recorded, has to be constant, and is given by the time resolution parameter t_res. Second, the instant at which the activity change has been introduced, has to be defined, which is done by the parameter t_inputchange. In the example given, the activity change was introduced between 270 and 271 s after the recording was started, thus t_inputchange is set to 270. Thereby tmin and tmax is defined, as well. As the response cannot occur before activity change the dead time refined by the algorithm is constrained to take positive values only.

In order to relate activity to the measured signal, the calibration parameters from the pH-meter are given. The algorithm can handle sensors with linear response curves (whether they are electrode or optode based). Implementation of response determination from sensors with non-linear response curves is possible, but one should always test the response in the operational range of the sensor, see Figure 12 and 13.

The model type is selected by the input ModelType, which take the numbers 1, 2, 3, 4 and 5 corresponding to the five models presented in this work (i)-(v), respectively. As the time constant k is different for each model, the parameter k_guess offers a possibility for controlling the start value of its refinement. Finally, two algorithm specific parameters are given: alpha_transient determines the transient region in which the second and final curve fitting is carried out. The transient region is defined by the conversion fraction alpha_transient. This parameter should never exceed αmax, which is determined from the S/N-ratio. The transient region is cut out for estimation of noise. ParTol defines the upper and lower boundary of the parameters that are re-determined in the second and final curve fit (activity step a0/a∞ and final signal S∞). These parameters should be well estimated already from the initial LTI-fit, so setting ParTol = 0.1 means that the parameters are allowed vary within 10 %, i.e. that they take values between 90 and 110% of the values first determined. Drift, xdrift, is fixed after the initial LTI refinement, as it is badly determined in the transient region. Slope, s, is fixed to the value calculated from the calibration parameters throughout the fitting. Dead time was constrained to refine within 0 and tres. In case larger dead time is observed, the setup has to be modified.

| S10

Result from data fitting

An example of how data should be reported is found in the paper. Here we report some additional comments to the results: The recorded response curve is shown in Figure S1. All five models presented in this work were used to fit a response curve to the data. As seen in Table S5, three models describe the transient response well: the hyperbolic, the boundary layer diffusion controlled and the bulk-optode diffusion controlled. The goodness of fit are equally good for the three models, and all three models agree on a 10-fold increased activity step change (a0/a∞ ~ 0.1), and a response time t90 of 3±1 s. The models were constrained to refine within a dead time of tres, which is 1 s. The validity of the setup was controlled and the input change was observed to be instantaneous with respect to the time resolution. The noise estimated by the different models are similar and in agreement with the resolution of the readout of the pH meter (±0.1 mV). The noise level corresponds to a maximum realistic conversion fraction of 99.6%. From the initial fit using the first-order LTI model, it was confirmed that steady-state measurements were achieved during the experiment. The boundary layer diffusion model was found to describe the transient response best, and a time constant kFD of 0.448 s–1 was determined. This corresponds to a response time t90 of 3.3 s.

Figure S1: Experimentally recorded response curves (blue) and simulated response curve (black) using the boundary layer diffusion controlled response model (iii), Eqn. (5). Activity step, steady state signal after activity change, drift and noise are determined mainly from the steady-state regions (left), the model type and response time are determined from the transient region (right).

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Table S5: Response time as determined from curve fitting.

Model

a0/a∞ S∞

(mV)

xdrift

(mV/s)

σ

(mV)

S/N kj

(s–1)

td

(s)

t90

(s)

t99

(s)

R2 SSE

(i) First-order LTI 0.10868 ± 0.00005

2.28 ± 0.01

-0.00045 ± 0.00002

0.21 271 0.326 ± 0.005

0 7.1 14.1 0.9975 1519

(ii) Hyperbolic 0.10662 ± 0.00005

2.77 ± 0.01

-0.00045 ± 0.00002

0.10 570 4.2 ± 0.1 0.75 ± 0.01

2.9 24.2 0.9997 154

(iii) Boundary layer diffusion

0.10875 ± 0.00005

2.26 ± 0.01

-0.00045 ± 0.00002

0.21 263 0.448 ± 0.008

0 3.3 8.3 0.9998 90

(iv) Bulk electrode membrane

0.10661 ± 0.00005

2.77 ± 0.01

-0.00045 ± 0.00002

0.15 376 33 ± 5 1 1.35 47.5 0.9972 1667

(v) Bulk optode diffusion 0.10875 ± 0.00005

2.26 ± 0.01

-0.00045 ± 0.00002

0.21 263 0.64 ± 0.02

0 3.3 6.9 0.9998 137

Activity step change, a0/a∞; steady-state signal after activity change, S∞; drift parameter, xdrift, given as the slope of a straight line fitted the data; standard deviation of

random error (i.e. noise), σ, from the measurement; signal to noise ratio, S/N; the time constant kj the jth model; dead time, td; response time tα given at conversion

fractions α of 90% and 99%, respectively; coefficient of determination R2 for the full data range; error sum of squares, SSE, for the full data range.

| S12

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