#  Determining the presence and the number of people by using a Wi-Fi signal

Dario Jukić<EMAIL_ADDRESS>Faculty of Civil Engineering, University of
Zagreb, A. Kačića Miošića 26, 10000 Zagreb, Croatia  Silvije Domazet
Department of Physics, Faculty of Science, University of Zagreb, Bijenička c.
32, 10000 Zagreb, Croatia  Ante Ivanko  David Raca  Siniša Nikolić  Marin
Knežević  Filip Jović  Nenad Raca  Aduro Idea, Zavrtnica 17, 10000 Zagreb,
Croatia.  Hrvoje Buljan  Department of Physics, Faculty of Science, University
of Zagreb, Bijenička c. 32, 10000 Zagreb, Croatia

###### Abstract

We present experimental results and theoretical methods for the precise
determination of the presence and the number of people in an observed area by
using Wi-Fi signals. Our setup does not require active cooperation of persons
present in the Wi-Fi field, and relies only on the received signal strength
indicator (RSSI), which is read by the detectors. We first show that the
standard deviation of the measured RSSI data can be used as a practical tool
to establish the presence of a person (or more persons) with high precision,
in particular when the signal source is inside the measurement room. For the
more difficult problem of counting the number of persons, we have employed
machine learning algorithms to analyze data collected on nine different
detectors and up to nine people present in our experiment. We have achieved
excellent results (prediction accuracy of $98\%$ and above) for counting
already with only few detectors utilized in the analysis.

##  I Introduction

The usage of Wi-Fi signals for establishing the presence of one or more
persons inside a room or an open space can be of great practical interest, in
particular when no active cooperation of persons is assumed (people do not
have to carry their own Wi-Fi devices). A more challenging problem is that of
counting the exact number of such persons, and to date it has been studied in
dominantly two different directions. The first approach for counting uses only
information about the signal strength on the detector, i.e,. the Received
Signal Strength Indicator (RSSI) value read by the detector. The so-called
Channel State Information (CSI) provides for an alternative route, as it
offers a much richer description of the signal, but is at the same time much
more complex in practical terms. We now first provide with a brief overview of
relevant works in both directions.

###  I.1 RSSI

The first studies (relevant to this topic) in the domain of radio frequencies
of electromagnetic radiation, which were based only on the RSSI value of the
detectors and with people who do not actively cooperate, have investigated the
level of presence (density) of people in an indoor environment [1], or
determined the exact number and location of small number of people (up to four
people in Ref. [2]).

Additional progress in determining the exact number of people using RSSI
signals was reported by Depatla _et al._ [3] (for patent application, see
[4]). Only one pair of antennas (transmitter and detector) was placed
stationary in the space, and there were people walking between them.
Mathematical analysis of two important effects by which people influence the
received signal strength was made. The first is blocking the direct line of
sight (LOS) when the individual is exactly on the transmitter-detector line,
and the second refers to the effects of signal scattering on people. The
result of this analysis is an expression for the probability distribution of
the received signal as a function of the number of persons present. By
comparing the theoretical model and experimental results, the authors
calculated the probability of success in counting people. For antennas
(radiating in all directions) in closed spaces, the accuracy of counting up to
a total of nine people was $55\%$ (with an error of up to one person), or
$63\%$ (with an error of up to two people). In the outer space, these
accuracies amounted to $64\%$ and $96\%$, respectively. Even better results
were achieved by using directional antennas. In their subsequent works, they
have also demonstrated the possibility of using RSSI data to determine the
average walking speed of people in space [5], and for counting people through
walls [6].

We highlight also the work [7] in which the authors collected the results of
RSSI measurements at ten measuring points (detectors) inside one room, and the
number of people was determined by using regression-based methods. In the
first case they used the linear, and in the second the so-called support
vector regression (SVR). SVR is a nonlinear regression method based on the
Support Vector Machine (SVM) algorithm, which is extremely popular in machine
learning for solving classification and regression problems. The authors used
these methods, among other things, to predict the presence of people in the
room (people are present or not), as well as to determine the exact number of
people. The SVR method proved to be more successful: in experiments with a
maximum of seven people in the room, the accuracy of predicting presence was
$98\%$, and determining the correct number of people was $77\%$.

###  I.2 CSI

A newer approach to determine the number of people in an environment uses CSI
data recorded by modern Wi-Fi systems. In this case, the Multiple Input -
Multiple Output (MIMO) systems in Wi-Fi technology enable the information
about the amplitude and phase of the signal to be recorded on many channels
(frequencies). The channels are orthogonal to each other thanks to the
Orthogonal Frequency Division Multiplexing (OFDM) technique, because the
frequencies are chosen in such a way as to avoid interference during
transmission (see, for example, Ref. [8]).

The first paper in which this approach has been used was published in 2014
[9]. Based on experiments (in which up to 30 people participated) and the Gray
Verhulst model, the authors determined the accuracy of predicting the number
of people. In indoor spaces, the accuracy was about $80\%$ (with an error of
up to one person), and $98\%$ (with an error of up to two people). In outdoor
conditions, the model was less successful, with an accuracy of about $70\%$
(with an error of up to two people). Subsequent papers have used different
methods, mainly using machine learning techniques, to predict the number of
people based on CSI data [10, 11, 12, 13, 14, 15]. In addition to determining
the number of people, the CSI offers many other potential applications, which
are described in more detail in the review article [16].

###  I.3 Summary of our findings

In this paper, we report on the measurements we have conducted and their
analysis, in order to find practical (efficient) algorithms for detecting the
presence and counting the number of people without their active cooperation,
based only on the RSSI values read by detectors. We show that the standard
deviation of the measured time series can be used as a feature which
discriminates between the presence of one (or more persons) and the noise in
an observed environment. Based on this conclusion, we define algorithms which
have been trained only on the noise data and have shown excellent prediction
accuracy for tested data samples, which is comparable to established (and more
demanding) machine learning algorithms. However, we emphasize that the
prediction quality can be somewhat reduced if the radiation source is located
outside the measurement area. In the second part of the work, we have studied
a more challenging problem of people counting, for this we have performed
measurements with nine detectors and up to nine people in the measurement
area. We have demonstrated that for this setup the well-known machine learning
algorithms (e.g., Random Forest) can be used for classification purpose with
excellent prediction results ($98\%$ and above), already when data from only
few (e.g., four) detectors are used.

###  I.4 Basic concept of experiments and their theoretical analysis

Our experiments and theoretical analysis have been conceived and carried out
as follows. One room for experiments has been planned, the Wi-Fi signal source
and Wi-Fi radiation detectors are located in that room. Alternatively, the
source (and even the detectors) could also be placed outside the measurement
room. Measurements were made by recording the RSSI values on the detectors.
For given positions of the source and detector, without the presence of people
or any other moving objects, the radiation intensity measured on the detectors
is constant with certain fluctuations (noise) which are a standard part of all
measurements. Noise analysis will prove extremely important for the
understanding and interpretation of measurement results.

When there are one or more people in the room, standing or moving, the
radiation intensity on the detectors will change in relation to the noise
value. Since the organs of the human body have a high dielectric permittivity
of the order of $50-80$ compared to vacuum at Wi-Fi frequencies [17], people
standing or moving scatter and absorb Wi-Fi radiation. These reflections and
absorptions cause a detectable change in radiation intensity over time. Based
on the collected data, i.e., time series of radiation intensity, the key part
of the work is to answer (a) whether someone was in the area where the
measurements were being taken, and (b) how many people were present? Various
methods can be used for this, from statistical analysis of measured time
series with and without people (and then building a predictive model), to
machine learning methods where the algorithm learns from previously measured
signals to make a prediction in real time. Furthermore, it may be useful to
examine with what reliability different methods work, what kind of algorithm
calibration is needed for a reliable result, and how the results change when
the source or detectors are moved to other positions. Which positions of
sources and detectors (and their number) are optimal for covering a certain
space? The answers to these questions will be discussed in the following
chapters.

##  II Detecting the presence of people: measurements

In this section we present two sets of experiments, both being setup with one
source and three detectors. However, the spatial configuration of the source
has been different: for the first set of measurements (labeled as M1) the
source was inside the room where the measurements have been made, while for
the the second set (labeled as M2) the source was placed outside. Schematic
illustration of the room and location of the source and the detectors for both
sets of experiments are shown in Fig. 1. We note that position of each
detector relative to the source will affect whether a given detector measures
more absorption or reflection of Wi-Fi radiation.

Figure 1: Floor plan of the measurement area and the position of the source
and detectors during (a) the first and (b) the second set of measurements. The
dimension of the measurement room is $3.5\times 4.5$ m2. (a) The source and
detectors are inside the room. (b) The source is moved outside the room, while
the positions of the detectors are not changed. Illustration of two processes
that lead to changes in the intensity of the measured signal: (i) reflection
and (ii) absorption by the human body. The signal in detector 1 is affected by
radiation absorption, while the signal in detector 3 is affected by radiation
reflection.

Let us consider the configuration of the source, detector and human body
graphically shown in Fig. 1(b). Here, the human body is between the source and
the detector 1 (D1); therefore, the measured radiation intensity at D1,
relative to the noise, will show that the radiation has been absorbed.
However, if we consider the positions of the source and the detector 3 (D3),
deviation from noise radiation on D3 will be caused by radiation reflection.

Obviously, this is a simplified picture of geometric optics that does not take
into account the wave nature of Wi-Fi radiation and possible interference from
radiation reflected from other objects. Detectors measure all possible
contributions and therefore, the analysis of measured signals also includes
all possible contributions. For an accurate theoretical description of the
system it is necessary to solve Maxwell’s equations in a given space with
adequate boundary conditions and an approximately accurate dielectric response
of the human body to Wi-Fi radiation. It should be noted here that a
simplified picture of absorption and reflection in geometric optics can still
give a quick answer on how to place sources and detectors in a given space to
maximize the desired precision of reading information from a Wi-Fi signal, and
is operationally better for use (it is significantly faster in time) than
solving Maxwell’s equations.

The sources and the detectors used in experiments operate at frequency of 2.4
GHz, and detectors can measure the radiation intensity up to a maximum of
about 200 measurements per seconds. However, we have verified that for our
purposes such a large number of measurements is not necessary, it has been
sufficient to have roughly an order of magnitude smaller measurement
frequency.

During the first set of measurements (M1, see Fig. 1(a)), we have recorded the
signal for 20 minutes on all three detectors, and the measurement conditions
did not change. That is, first we recorded the noise - no one was in the room
or near the room. After that, there was one person in the room and the signal
was recorded for the next 20 minutes; the person moved randomly (not too fast
but not uniformly), the person could also stop or make various movements.
After that, we have recorded more measurements for 2, 3, 4 and 5 people inside
the room. In this way, we have obtained 6 time series of 20 minutes each
(approx. 200 measurements per second) for each detector. We have also repeated
the measurements the following day. For the second set of measurements (M2),
the configuration of the system was changed (see Fig. 1(b)), and the
measurements were taken in an analogous way as described for the M1.

##  III Detecting the presence of people: analysis

In real systems, it is often the case that we have an imbalance in the amount
of data, i.e. much more data of only one type (e.g. noise) compared to others.
In that case, we can consider the majority of the data as normal system
behavior, and the other data as anomalies. Since we have few anomalies, and
they can generally be of different shapes, a useful approach is to look for
algorithms that will be trained exclusively on data that describe the normal
state of the system (in our case, noise), i.e., without the presence of
anomalies. The key task is to determine whether someone was in the room during
the tested time interval. The classification label for the detection of
presence is a binary variable: 0 – noise (no people), or 1 – there is someone
in the room.

It has already been clear to us, from several small preliminary signal
samples, that signal fluctuations depend in some way on the presence of people
in the room. For this reason, we have studied the moments of the distribution
of all signals, that is, we have calculated the statistical indicators of the
signal: mean value, standard deviation, skewness, kurtosis; and also the
Fourier spectrum. When doing such analysis, it was necessary to decide how
long each studied time series should be, given that the Fourier spectrum and
other parameters oscillate somewhat in time even when the signal recording
conditions do not change. We have been guided by the idea that within 10 or 20
seconds one would like to know the number of people in the room, or whether
there is someone in the measurement area or not. However, it is possible to
imagine applications where it is sufficient to know how many people have
stayed in a certain space for a long time interval on average.

Let us imagine that from one measured time series of length $T=20$ min we want
to extract quantity $A$ (eg., standard deviation, etc.) by the following
procedure. Each time series is split into shorter time series of $\tau$
seconds. Each method, depending on the goal of the application, has a
corresponding optimal value of $\tau$ (in what follows, we set $\tau=20$ s).
In this way, from one series we get $N_{\tau}=\text{floor}(T/\tau)$ shorter
time series ($(T/\tau)$ is rounded to the nearest whole number towards zero).
Then, for each series of $\tau$ seconds, we calculate the value of $A$; we can
do this for each detector separately. (However, we note that it is also
possible to first adequately combine the data from different detectors (eg. by
multiplication) and then calculate the magnitude of $A$ for the combined
series.)

###  III.1 Information extraction from statistical parameters

To begin with, we analyze moments of distribution for the measured signals in
order to develop appropriate algorithms for given applications. For the first
set of measurements (M1), each time series of $20$ minutes is split into
shorter time series of $\tau=20$ seconds. In Fig. 2 we show dependence of the
mean value of the signal and the standard deviation on the number of people.
It is clear from the figure that the standard deviation is correlated with the
presence of people and can be used as a discriminating factor in decision-
making. On the other hand, the mean value seems not to be correlated and its
relevance is questionable.

Figure 2: Mean value (a) and standard deviation (b) of the RSSI signal as a
function of the number of people in the room covered by the Wi-Fi signal.
Noise is marked with the number 0. Total length of the measurement is 20
minutes for each number of people, and the signal is split into series of
20-second samples. The mean value and standard deviation were calculated and
plotted for the 20-second signal samples. There is no correlation between the
mean value and the presence of people; however, a correlation is observed
between the deviation and the presence of people.

We interpret this observation as follows: if people are in the room and are
moving, the radiation is reflected from them and causes larger signal
fluctuations which are seen in the standard deviation of the signal. That is,
we find that the values of standard deviation for the noise measurements (0
people) are significantly smaller than its values when people were present in
the room. Therefore, we conclude that (for this configuration) the standard
deviation could be used to detect the presence of people. The advantage of
this method is that it is not necessary to calibrate (train) the predictive
model with measurements when people are present, but only with noise
measurements. We have validated this conclusion for all three detectors used.
In addition, we note that the standard deviation cannot be used (as the only
parameter) to extract precise information about the number of people in the
room. (We will later discuss this problem with machine learning approaches.)

On the other hand, the mean value is affected by the position of the person in
such a way that the signal value increases through reflection and decreases
through absorption, so that there is no one-way influence of the presence of
people in the room on the mean value of the signal. We have also calculated
other statistical quantities, i.e., the skewness and the kurtosis (not shown).
We had noted that noise values for these parameters are more scattered when
compared to the signal with people present, however, there is no clear
discriminating procedure by which these parameters would be used for the
binary decision-making application (i.e., to decide on presence or absence of
people). Therefore, we do not focus on these parameters in the present
discussion.

####  III.1.1 Predictive model (algorithm) for binary classification based on
standard deviation of the noise - Method 1

The input data for the algorithm are only the noise measurements. The (average
value of) standard deviation $\langle\sigma\rangle$ has been extracted for all
three detectors (we focus first on M1 configuration):
$\langle\sigma\rangle_{D1}=0.4317$, $\langle\sigma\rangle_{D2}=0.4214$, and
$\langle\sigma\rangle_{D3}=0.4787$.

From Fig. 2 it is clear that if the standard deviation (of a 20-seconds time
series) is greater than a factor $f\cdot\langle\sigma\rangle$, then there is
almost certainly someone in the room. Now it is necessary to determine the
optimal value for the factor $f$. Based on our measurements, we have chosen
the value $f=2.2$. If the detector D$j$ ($j=1,2,3$), within $\tau=20$ s, has a
standard deviation greater than $2.2\langle\sigma\rangle_{Dj}$, that detector
indicates that someone is in the room. Otherwise, there is no one in the room.
Since we have three detectors, it is necessary to make a final decision from
their output results. We have given every detector the same weight in the
decision. That is, if at least two detectors suggest that someone is in the
room, the decision is that someone is in the room.

We tested the algorithm on measurements from the M1 set (with 1 to 5 people;
and with 0 to 4 people, taken the following day). Prediction accuracy of the
algorithm has been $100\%$ for all measurements, with a single exception of
one measurement with one person present and recorded one day after the noise
measurement used for training, for which we found $90\%$ accuracy.

Let us focus now only on that weaker result. Fig. 3 shows the algorithm
predictions for that measurement (with blue line for the method we presently
discuss). There is only one longer time interval of about 1 minute when the
method missed the correct value. We conclude that even for that worst
measurement, the algorithm can predict within a longer time interval whether
someone is in the room. Furthermore, it is necessary to take into account the
long-term stability of the algorithm, i.e, examine whether the algorithm
becomes worse if it is not recalibrated for days.

Figure 3: Prediction of two algorithms for binary classification (blue: Method
1, red: Isolation Forest) for the measurement made one day after the
calibration and with one person present in the measurement room ($1$ stands
for the presence being detected, and $0$ for the noise). This was the only
measurement in the M1 set (configuration with both source and detectors inside
the room) for which the algorithms could not reach $100\%$ accuracy (here:
$90\%$ for Method 1, and $93\%$ for Isolation Forest).

We have also trained and tested the algorithm on measurements from the M2 set.
From Table 1 we can see that the results are somewhat worse than for M1, which
we associate with a different position of the source (see Fig. 1). However,
they were satisfactory for all measurements, except for the M-211 (one person
present).

Measurement |  M-211 |  M-212 |  M-213 |  M-214 |  M-22X |  M-22Y |  M-22Z
---|---|---|---|---|---|---|---
Number of persons |  1 |  2 |  3 |  4 |  1 |  2 |  3
Method 1 |  $45\%$ |  $88\%$ |  $95\%$ |  $100\%$ |  $83\%$ |  $97\%$ |  $100\%$
Method 2a |  $72\%$ |  $98\%$ |  $100\%$ |  $100\%$ |  $100\%$ |  $98\%$ |  $100\%$
Method 2b |  $67\%$ |  $97\%$ |  $100\%$ |  $100\%$ |  $100\%$ |  $98\%$ |  $98\%$
Isolation Forest |  $72\%$ |  $97\%$ |  $100\%$ |  $100\%$ |  $100\%$ |  $100\%$ |  $100\%$
Table 1:  Prediction accuracy of four different algorithms used for presence
detection for the measurement set M2; detection is more challenging since the
radiation source is placed outside of the measurement room.

Our conclusion on the usability of the algorithm is as follows. This algorithm
can be used for practical purposes for precise detection of whether or not
there are people in the Wi-Fi signal field. The precision of the algorithm can
be extremely high, but it depends on the position of the source and the
detector. Importantly, in longer time intervals (for example, averaged results
for 1-2 minutes), the algorithm practically makes no errors.

The notion that standard deviation is the statistical parameter which is well
correlated with the presence (or absence) of people can be implemented in
several other ways. In what follows we provide two specific approaches.

####  III.1.2 Predictive model (algorithm) for binary classification - Method
2a

We have already seen that the noise deviation is significantly smaller than
the signal deviation when people are present in a measurement area. By
multiplying the deviations at different detectors, such differences can become
even larger (more pronounced). For each detector, we determine standard
deviation of the total signal (for the whole measurement of 20 minutes) and
then define the product of these deviations as the correlated noise deviation
(for three detectors, we multiply three deviations). After that, we analyze
the test signal (of length $\tau=20$ s): we find the standard deviation for
each detector, and their product defines the correlated deviation. A simple
protocol for decision-making can be defined as follows: if the correlated
deviation of the test data is by a factor 3 (or more) greater than the
correlated noise deviation, then the algorithm predicts that (one or more)
persons are present in the room. Otherwise, there are no people in the room.

####  III.1.3 Predictive model (algorithm) for binary classification - Method
2b

We first multiply noise measurements from all three detectors at the same time
(during the entire period, in our case 20 minutes) and find the standard
deviation of such data set ($\sigma_{0,D1D2D3}$). For testing data (of length
$\tau=20$ s) we calculate the correlated data by multiplying the measurements
from all 3 detectors at the same time. Protocol for decision-making is defined
as follows: if the standard deviation of the tested correlated data is by a
factor 3 (or more) greater than $\sigma_{0,D1D2D3}$, the algorithm says that
(one or more) persons are present in the room. Otherwise, there are no people
in the room.

###  III.2 Information extraction using machine learning

Finally, we have conducted the analysis of our measurements with machine
learning techniques. We have extracted features which characterize individual
time series using the readily available software packages [18], and then
treated them as input variables for the classification problem. We have
employed several well-known algorithms which can be used for semi-supervised
learning (when only the noise data is used for algorithm training), and have
chosen the Isolation Forest algorithm to be the well suitable for our data
sets (other possible choices include One-Class Support Vector Machine, Local
Outlier Factor, Elliptic Envelope, etc.) This approach can also serve as a
benchmark for the quality of previously introduced statistical methods.

###  III.3 Comparison of different methods

For the source and detector configuration of M1 data sets (when both source
and detectors are inside the measurement room), all algorithms give
approximately equal results, and they are extremely precise. Specifically,
prediction scores have not been perfect (but still of the order of $90\%$ or
more) in a single case: when one person was inside the room and during the
measurement taken one day after the noise measurements used for calibration
(algorithm training), see Fig. 3. We conclude that the sensitivity (i.e.,
precision) can change over time, so it is necessary to calibrate the measuring
system (parameters used by the algorithms) every day (calibration can be
preferably done at night when there is no one in the room).

For the M2 configuration (when the radiation source is placed outside the
room), the prediction results of all algorithms are generally somewhat worse
than for the M1, and they are summarized in Table 1. (Numbers presented may
marginally change depending on data sampling rate, the choice of time series
length $\tau$, or machine learning (for the Isolation Forest) parameters.)
Notably, algorithms are of limited use in the M2 configuration when only a
single person is present in the measurement room (in particular, Method 1
gives worse results on measurements).

Obviously, we find that the precision of the measurements (and the prediction
accuracy of algorithms) depends on the configuration of the source and the
detectors. (Throughout this discussion we have relied on data from three
detectors, but in real-life situations even a single detector may be
sufficient.) Finally, we conclude that with the daily noise calibration, it is
possible to employ practical algorithms that will precisely detect people
presence in the measurement field, by using only the RSSI signal. Algorithms
have correlated errors, so unfortunately they cannot be used as vectors in the
sense that when one fails we use the other one.

##  IV Counting the number of people: Measurements and analysis

In the second part of the research, we have aimed at finding an optimal
approach to the more complicated problem of counting people based on the
measurements of Wi-Fi signal strength. It is important to emphasize that for
this analysis we have conducted experiments wither larger number of detectors
than in the first part, in order to generate more accurate predictive models.

The geometry of the experimental setup (with radiation source and detectors)
is presented in Fig. 4.

Figure 4: Floor plan of the experimental setup for Wi-Fi signal measurements
with one source and 9 detectors. The source (and detectors) were placed at a
height of 0.5 (and 1) m above the ground. The dimension of the measurement
room is $4\times 4.5$ m2.

We had a total of nine detectors placed in different places within the room,
which recorded Wi-Fi signal strength (RSSI) over time. We have made
measurements in the presence of one, three, five, seven or nine people. People
were either at rest or in motion. For each cycle with a defined number of
people, we collected data for twenty minutes: ten minutes with people standing
still and ten minutes when people are moving. We also collected the noise
data, without people present, in order to have reference data on the signal
strength in the room.

We first divided the time series obtained by measurements into intervals of 20
seconds. In this way, for each measurement that lasted 20 minutes, we obtained
a total of 60 different subintervals. Considering that we had a total of 5
such measurements (for the number of persons 1, 3, 5, 7 or 9), we finally had
a set of 300 different time intervals at our disposal.

As already established, determining the number of people in the observed room
based on the strength of the electromagnetic (Wi-Fi) signal can be defined as
a classification problem in the field of machine learning (here, the
classification label is the number of people in the room, and we will discuss
supervised learning algorithms). For each of the 300 time series, we have
performed the extraction of relevant featured that characterize it (as noted,
we did this using a software package [18]); these features then represent
input for specific algorithms. In accordance with the standard approach,
collected data can be divided into two sets, for training and for testing of
the algorithm. As a result, we get predictions of classification labels, i.e.,
the predicted number of people in the measurement room during the tested time
interval.

We have made the division into a training set and a testing set by using
cross-validation: this means that the data is divided into $k$ subsets, where
$(k-1)$ subsets are used for training and the remaining one for testing. Our
set consists of a total of 300 elements, and for this analysis we have chosen
the value of the cross-validation parameter $k=3$, which meant that the tested
samples represent one third of the available data. In this way, training of
each algorithm (and then its testing) can be carried out three times (for
convenience, we label them as Run 1, Run 2 and Run 3).

The Table 2 shows the prediction success of three well established
classification algorithms, based on measurements data from all 9 detectors.

Algorithm | KNeighbors | Decision Tree | Random Forest
---|---|---|---
Run 1 | $95\%$ | $92\%$ | $99\%$
Run 2 | $96\%$ | $91\%$ | $98\%$
Run 3 | $93\%$ | $97\%$ | $100\%$
Average | $95\%$ | $93\%$ | $99\%$
Table 2:  Prediction accuracy of three different machine learning algorithms
(K-nearest neighbors, Decision Tree, and Random Forest) for classification
problem of determining the number of people in the room, and for the test data
recorded on nine detectors. Algorithms have been employed with cross-
validation parameter $k=3$ (2/3 of data for training, 1/3 for testing),
resulting in three separate runs (models).

We note that all three selected algorithms are excellent in predicting the
number of people in the room: K-Nearest Neighbors (KNeighbors), Decision Tree,
and Random Forest (it also seems that this one should be highlighted in terms
of success). We have also made preliminary tests with neural network
algorithms (not shown here); however, this approach could be more useful (and
successful) with larger data sets which we do not presently have (and further
optimized by changing the parameters of the neural network).

Given that the prediction results based on the data of all 9 detectors are
excellent, below we will demonstrate how these results depend on the number of
detectors. The goal of this check is to determine the minimum number of
detectors in the room required to obtain satisfactory prediction accuracy.
Fig. 5 shows prediction accuracy of the chosen algorithms as a function of the
number of detectors used in the analysis. It can be seen that the prediction
success initially increases with the increase in the number of detectors, but
that already at the number of 4 detectors, successful algorithms reach
excellent prediction power, which changes only marginally with an additional
increase in the number of detectors. Most notably, the Random Forest has
performed with accuracy of $98\%$ when four detectors (labeled as 1 to 4 on
Fig. 4) are used, and reaches $99\%$ already with five detectors (1 to 5 on
Fig. 4). Obviously, these results may slightly change depending on the spatial
configuration of the detectors used inside the measurement area.

Figure 5: Prediction accuracy of three different algorithms (K-nearest
neighbors, Decision Tree, and Random Forest) for the classification problem of
counting the number of people inside the measurement room, plotted as a
function of the number of detectors used for analysis.

Finally, we emphasize here the most important results in this part of the
research. Machine learning algorithms have shown an excellent ability to
predict the number of people inside the measurement room based on the Wi-Fi
(RSSI) signal, in an experimental setup with up to 9 people (and a total of 9
detectors). In particular, we have shown that the algorithms can achieve
exceptional prediction success when analyzing experimental data even with only
few detectors present in the room. For example, it is worth noting that the
Random Forest algorithm performed with $98\%$ ($99\%$) success rate for the
data collected on four (five) detectors for our classification problem. As an
outlook, this study could be extended to an even larger experimental setup,
with more people (and possibly more detectors); exact counting becomes more
demanding, but finding an optimal (and cost-effective) approach will surely
depend on specific applications.

##  V Conclusion

In summary, we have measured the strength of the Wi-Fi signal (i.e., the RSSI
value) on various detectors with the aim to determine the presence (or
absence) and the number of people in an observed environment. Importantly,
persons involved do not actively cooperate (signal strength on their Wi-Fi
devices is not recorded). Our subsequent analysis has shown that the presence
of one (or more) persons can be established with only a single quantity: the
standard deviation of the RSSI value, and a small number of detectors (up to
three in our experiment). This understanding can be implemented in several
ways, all of which lead to decision-making algorithms which can be of high
precision when making a binary prediction (presence or absence). They compare
well with more advanced methods (such as machine learning algorithms), yet
remain relatively simple. However, we have also established that the same
algorithms can be of limited use in experimental setup when the source of the
Wi-Fi signal is located outside the observed environment.

As for the more challenging problem of counting the number of persons, more
detectors are generally needed (we had a total of nine detectors placed in an
indoor environment) and the data analysis has been performed by employing
standard machine learning algorithms (a single feature, such as the standard
deviation of the RSSI, does not perform well in this classification problem).
For our experimental setup (with up to nine people), we have verified that
algorithms were counting the number of people with high accuracy. For example,
the Random Forest algorithm has had the average success rate of $99\%$ (the
classification problem was designed to discriminate between labels 1, 3, 5, 7,
and 9 for the number of people). Importantly, we have also shown that the
excellent prediction accuracy of ML algorithms can be achieved when taking
into account data from only few detectors. This final observation has also
been motivated by possible real-life practical applications, where employing a
minimal yet accurate system, is very beneficial.

##  VI Acknowledgment

The work was supported in part by the European Union through the European
Regional Development Fund (Grant No. KK.01.2.1.02.0016). The authors would
like to thank all colleagues, either at the University of Zagreb or at the
Aduro Idea, who helped in organization of the experiments and with their
presence when the experimental data were being taken.

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# Accuracy and speed of elongation in a minimal model of DNA replication

M Sahoo<EMAIL_ADDRESS>Department of Physics, University of Kerala,
Kariavattom Campus-6955881, India  School of Physics, Indian Institute of
Science Education and Research, Thiruvananthapuram-695551, India  Arsha N
Department of Physics, University of Kerala, Kariavattom Campus-6955881, India
P R Baral  School of Physics, Indian Institute of Science Education and
Research, Thiruvananthapuram-695551, India  S Klumpp<EMAIL_ADDRESS>goettingen.de Institute for the Dynamics of Complex Systems, University of
Göttingen, Göttingen, Germany

###### Abstract

Being a dual purpose enzyme, the DNA polymerase is responsible for elongation
of the newly formed DNA strand as well as cleaving the erroneous growth in
case of a misincorporation. The efficiency of replication depends on the
coordination of the polymerization and exonuclease activity of DNA polymerase.
Here we propose and analyze a minimal kinetic model of DNA replication and
determine exact expressions for the velocity of elongation and the accuracy of
replication. We first analyze the case without exonuclease activity. In that
case, accuracy is determined by a kinetic competition between stepping and
unbinding, with discrimination between correct and incorrect nucleotides in
both transitions. We then include exonuclease activity and ask how different
modes of additional discrimination in the exonuclease pathway can improve the
accuracy while limiting the detrimental effect of exonuclase on the speed of
replication. In this way, we ask how the kinetic parameters of the model have
to be set to coordinate the two activities of the enzyme for high accuracy and
high speed. The analysis also shows that the design of a replication system
does not universally have to follow the speed-accuracy trade-off rule,
although it does in the biologically realized parameter range. The accuracy of
the process is mainly controlled by the crucial role of stepping after
erroneous incorporation, which has impact on both polymerase and exonuclease
activities of DNA polymerase.

###### pacs:

75.50.Ee, 71.20.Ps, 75.10.Pq, 75.30.Kz, 75.30.Et, 75.10.Jm

##  I Introduction

The genetic information of a cell or an organism is stored in its DNA
(deoxyribonucleic acid) and transmitted to the next generation through the
process of DNA replication [1, 2]. DNA replication is a complex process that
involves multiple enzymes including helicases, nucleases and topoisomerases
on[3, 4, 5, 6, 7]. At its core, however, is DNA polymerase (DNAP), which
catalyzes the actual copying of the genetic information [8]. During the
replication process, the DNAP acts as a molecular motor and moves step by step
on the template DNA strand converting chemical energy into directed motion [9,
10, 11]. At the same time it copies the genetic information of the DNA
sequence along which it moves into a newly synthesized DNA molecule with
complementary sequence following the Watson-Crick complementary base pairing
rule[2]. The movement of DNAP on the template occurs in stochastic deoxy-
nucleotide triphospate (dNTP) base steps. One single base step of the DNAP
corresponds to a single-nucleotide elongation of the new DNA strand.

DNAP acts as a dual-purpose enzyme having polymerase as well as exonuclease
activity during the replication process [12, 13, 14, 15]. While its primary
function is to elongate the newly formed DNA strand, it can also cleave the
newly formed DNA strand and remove the last nucleotide when this has been
transfered to the exonuclease site of the enzyme. This activity functions as a
proofreading mechanism: Upon misincorporation (incorporation of a wrong dNTP),
the DNAP switches its functionality and the new strand is transferred to the
exonuclease site where the wrong nucleotide is cleaved from the new strand
through hydrolysis [16], before the DNA strand returns to the active site for
polymerization for elongation to continue [14]. This mechanism represents an
intrinsic proof-reading mechanism known as exonucleolytic proofreading [17, 1,
18, 2, 19, 16] as misincorporated nucleotides are excised, thus resetting the
incorporation process, such that the errors can be corrected by a second
attempt at correct incorporation. Polymerase and exonuclease sites of the DNAP
are separated from each other by a distance of 3-4 nm [19, 20, 21, 22], and
the transfer of the new DNA strand between these two sites may involve one or
many intermediates [23, 24, 25]. The dynamics of the transfer reaction is not
understood in detail.

Just like wrong nucleotides can erroneously get incorporated during
polymerization, compromising the fidelity of replication [26], correctly
incorporated nucleotides can erroneously get transferred to the exonuclease
site and be cleaved, resetting the incorporation process to the start and
hence compromising the speed of elongation.. Thus, the coordinated action of
the polymerase and exonuclease activity of DNAP is crucial for an effective
and faithful replication mechanism and determined the velocity and accuracy
are the two fundamental parameters in deciding the speed as well as the
fidelity of replication.

It is usually argued that an enhancement of accuracy results in a slow down of
the elongation process, resulting in a speed-accuracy trade-off [27, 28, 29,
6]. The high fidelity or enhancement of accuracy is believed to be as a matter
of proof-reading mechanism during the exonuclease activity [17, 30]. However,
a recent study [31] claims that the speed-accuracy trade-off is not universal,
but rather depends on the kinetic parameters of the enzyme. In this work we
study a simple kinetic model of DNA replication, for which we can exactly
calculate the velocity and accuracy of replication to investigate the
interplay of the two functions of DNAP. Based on that analysis we compare
different possible schemes of discrimination between correct and incorrect
nucleotides and address how an accurate replication mechanism can be achieved,
while also limiting the negative effect on the speed of elongation.

Figure 1: Minimal model for the dynamics of DNA polymerase (DNAP). Left panel:
Schematic depiction of the replication process with the leading DNA strand
passing through the DNA polymerase (encircled area). Right panel: Kinetic
model for DNA replication. Three sites on the DNA are labelled as $i-1$, $i$,
and $i+1$ and for the $i^{\mathrm{th}}$ site, the states of the DNAP are
labelled as $A$, $C$, $IC$, $C_{EX}$, and $IC_{EX}$, respectively. Binding of
a correct nucleotide takes the DNAP to the state $C$, whereas binding of an
incorrect nucleotide takes the DNAP to the state $IC$. Actively elongating
DNAP takes a forward step with rate $\epsilon$ from the state $C$ and with
rate $\epsilon^{{}^{\prime}}$ from the state $IC$. Exonuclease activity is
described by the states $C_{EX}$ and $IC_{EX}$, depending on whether it is
initiated from a correct or from an incorrect nucleotide. The arrows indicate
the transition from one state to the other state with their corresponding
transition rates as mentioned. The bond marked in red in state $IC$
corresponds to the nucleotide mismatch and thus to a weaker bond in
state-$IC_{EX}$.

##  II Model

###  II.1 Model with parallel pathways for correct and incorrect dNTP
incorporation

To study the interplay of polymerization and exonuclease activity, we use a
minimal stochastic model, which is depicted in Figure 1 as a schematic
diagram. The DNA template is considered as an one-dimensional lattice of
certain length and each site of that lattice represents an individual base of
the DNA template. The DNA polymerase (DNAP) moves on the DNA template by
single base steps, simultaneously extending the copy by one deoxy-nucleotide
triphosphate (dNTP) in each step. As a minimal description, we describe this
process by Michaelis-Menten kinetics, reversible binding of the dNTP followed
by irreversible incorporation into the new DNA strand. We note that this
kinetics have been characterized in more detail and exhibit additional
substeps, but here we aim at a minimal model. In the stochastic model, these
two substeps are represented by transitions from the active state with a free
binding site ($A$) to the occupied state (denoted by $C$) and back, where
there are two pathways for the enzyme to transition back to the free state,
either by unbinding of the dNTP or by incorporation and stepping, in which
case the active and free state is reached at the subsequent position and with
a one based longer new DNA strand. The rates of the three transitions are
denoted as $k_{c}$, $k_{-c}$ and $\epsilon$, respectively.

To account for misincorporations and thus for errors in replication, the model
includes a second pathway with Michaelis-Menten kinetics that describes
exactly the same steps as the previous one, however for binding and
incorporation of an incorrect, i.e. non-cognate dNTP. The corresponding
occupied state is denoted by $IC$ and the rates, which may generally differ
from those in the correct pathway, with $k_{ic}$, $k_{-ic}$ and
$\epsilon^{\prime}$, respectively.

###  II.2 Proofreading pathway

This basic model is extended to include the exonuclease activity of DNA
polymerase. For that, the polymerase transitions to a state with the newly
added dNTP in the exonuclease site. The exonuclease state is denoted by
$C_{EX}$ or state-$C_{IEX}$, depending on whether the new nucleotide is a
correct or an incorrect one. The transition to that state occurs from state-C
(or state-IC) with rate $k_{\mathrm{ex}}$ (or $k_{\mathrm{iex}}$),
respectively. The reverse transition is also included in the model (with rates
$k_{\mathrm{-ex}}$ and $k_{\mathrm{-iex}}$). Finally, the exonuclease reaction
occurs from state $C_{EX}$ or state-$C_{IEX}$ with the cleavage rate
$k_{\mathrm{cl}}$ or $k_{\mathrm{cl}}^{{}^{\prime}}$, respectively. In this
transition, the last incorporated nucleotide is cleaved off and the enzyme
goes back to the free state $A$.

We note that the reaction scheme described here differs from the classical
kinetic proofreading scheme [3], as the proofreading step requires a
transition away from the main pathway, namely the transfer of the new DNA
strand from the polymerization site to the exonuclease site. A reaction scheme
that is very similar to ours (and also describes the off-pathway proofreading
step) has been used by Sharma and Chowdhury [13]. Their focus however was on
the stochasticity of DNA elongation by calculating dwell time distributions
rather than on the different mechanisms of discrimination between the correct
and incorrect nucleotides and on how they affect the design of accurate
replication and the speed-accuracy trade-off. There are also some smaller
differences between these two models. In their model, they include a
transition to the exonuclease site before incorporation, which does not affect
the accuracy (and is not included here), and do not explicitly represent the
correct and incorrect incorporations as parallel pathways. We also note that a
related, but considerably more complex proofreading scheme is found in RNA
polymerase: RNA polymerase transitions to an off-pathway state for
proofreading, the backtracking state, where it however performs diffusive
motion and can cleave longer pieces of the newly synthesized RNA [32, 35].

For discrimination between correct and incorrect nucleotides, we generally
expect the following relations to hold: In the initial binding step, binding
of incorrect nucleotides should be suppressed compared to binding of correct
nuclotides. This may be due to a lower binding rate and/or a higher unbinding
rate, reflecting a higher affinity for correct than for incorrect binding. The
discrimination achieved in this step is restricted thermodynamically, by the
difference in binding energy between the correct and incorrect dNTPs [3].
Additional discrimination can be expected based on the irreversible step
(translocation of the enzyme), typically with $\epsilon^{\prime}<\epsilon$. In
contrast to binding, discrimination by the translocation rate is kinetic
rather than thermodynamic and thus is not restricted by the binding energies
but rather by the difference in the rates or in the corresponding energy
barriers. Finally, the proofreading pathway is expected to also contribute
kinetically to the accuracy, as transfer to the exonuclease site competes
kinetically with the translocation of the enzyme. If the transfer for
incorrect nucleotides is faster than that for correct nucleotides (i.e., for
$k_{\mathrm{i}ex}>k_{\mathrm{e}x}$), while at the same time if the
translocation is slower ($\epsilon^{\prime}<\epsilon$), one can expect the
typical fate of an incorrect nucleotide to be excision and that of a correct
one translocation. This tendency will be further enhanced if for correct
nucleotides in the exonucleotide site, return to the polymerization site is
more likely than cleavage, $k_{\mathrm{-ex}}>k_{\mathrm{cl}}$, while it is
just the opposite for the incorrect nucleotides in the exonuclease site,
$k_{\mathrm{-iex}}<k_{\mathrm{cl}^{\prime}}$. Estimates of the rates from the
literature are listed in Table I. Note that in some cases, these are estimated
from more detailed kinetic schemes. All rates in this table are scaled with
respect to the forward stepping rate $\epsilon$.

Table 1: Estimated values of the model parameters Parameter |  Symbol |  Value |  Refs
---|---|---|---
Forward stepping rate |  $\epsilon$ |  $300s^{-1}$ |  33, 34, 12, 8
Binding rate (correct dNTP) |  $k_{c}$ |  $10^{-1}\epsilon$ |  34
Binding rate (incorrect dNTP) |  $k_{ic}$ |  $10^{-2}-10^{-1}\epsilon$ |  15
Unbinding rate (correct dNTP) |  $k_{-c}$ |  $10^{-2}\epsilon$ |  15
Unbinding rate (incorrect dNTP) |  $k_{-ic}$ |  $10^{-2}-10^{-1}\epsilon$ |  15, 3
Erroneous stepping rate |  $\epsilon^{{}^{\prime}}$ |  $10^{-3}-10^{-2}\varepsilon$ |  33, 34, 8
Exonuclease site transfer rate (correct nucleotide) |  $k_{ex}$ |  $10^{-3}\varepsilon$ |  34
Reverse transfer rate (correct nucleotide) |  $k_{-ex}$ |  $<700s^{-1}$ |  33, further comments
Exonuclease site transfer rate (incorrect nucleotide) |  $k_{iex}$ |  $10*k_{ex}$ |  34
Reverse transfer rate (incorrect nucleotide) |  $k_{-iex}$ |  $10^{-4}\varepsilon$ |  34
Cleavage rate (correct nucleotide) |  $k_{cl}$ |  $10^{-2}\varepsilon$ |  34
Cleavage rate (incorrect nucleotide) |  $k_{cl}^{{}^{\prime}}$ |  $10^{-1}\varepsilon$ |  3

###  II.3 Steady state solution

In the following, we will analyze the elongation speed and the accuracy of
replication for two cases. In the first case, we consider a purely
polymerizing scenario, where the DNA polymerase does not undergo the
transition to the exonuclease state (i.e.,
$k_{\mathrm{ex}}=k_{\mathrm{iex}}=0$). In the second case, we include the
exonuclease activity. In both cases, we calculate the accuracy and velocity of
elongation from the steady-state fluxes between the states of the discrete
stochastic model described so far. The approach is similar to that used in our
earlier work for backtracking of RNA polymerase [35] and provides the mean
values of these quantities. Consideration of the fluctuations as in Ref. [13]
requires time dependent methods. Effectively, our system is a stochastic
5-state model. In the construction of our model, we have assumed that there is
no sequence dependence of the rates [4, 36, 37], so all the rates considered
here should be considered averages over a possible sequence-dependence. To
determine steady-state fluxes, we solve the five state model shown in Fig. 1
and first determine the steady state probabilities of the five different
states of the DNAP. These probabilities are denoted by $P_{\mathrm{i}}$, where
$i=\mathrm{A}$ (active state), $\mathrm{C}$ (correct state), $\mathrm{IC}$
(incorrect state), $\mathrm{C_{\mathrm{EX}}}$ (correct exonuclease state), and
$\mathrm{I_{\mathrm{CEX}}}$ (incorrect exonuclease state), respectively. We
calculate the steady state probabilities by equating the total incoming flux
and the total outgoing flux for each state, which can be expressed as

$k_{\mathrm{c}}P_{\mathrm{A}}+k_{\mathrm{-ex}}P_{\mathrm{C_{\mathrm{EX}}}}-\epsilon
P_{\mathrm{C}}-k_{\mathrm{ex}}P_{\mathrm{C}}-k_{\mathrm{-c}}P_{\mathrm{C}}=0,$
(1)

$k_{\mathrm{ic}}P_{\mathrm{A}}+k_{\mathrm{-iex}}P_{\mathrm{I_{\mathrm{CEX}}}}-\epsilon^{{}^{\prime}}P_{\mathrm{I_{\mathrm{C}}}}-k_{\mathrm{iex}}P_{\mathrm{I_{\mathrm{C}}}}-k_{\mathrm{-ic}}P_{\mathrm{I_{\mathrm{C}}}}=0,$
(2)

$k_{\mathrm{ex}}P_{\mathrm{C}}-k_{\mathrm{cl}}P_{\mathrm{C_{\mathrm{EX}}}}-k_{\mathrm{-ex}}P_{\mathrm{C_{\mathrm{EX}}}}=0,$
(3)

and

$k_{\mathrm{iex}}P_{\mathrm{I_{\mathrm{C}}}}-k_{\mathrm{cl}}^{{}^{\prime}}P_{\mathrm{I_{\mathrm{CEX}}}}-k_{\mathrm{-iex}}P_{\mathrm{I_{\mathrm{CEX}}}}=0.$
(4)

Together with the normalization condition for the probabilities,

$P_{\mathrm{A}}+P_{\mathrm{C}}+P_{\mathrm{I_{\mathrm{C}}}}+P_{\mathrm{C_{\mathrm{EX}}}}+P_{\mathrm{I_{\mathrm{CEX}}}}=1,$
(5)

the steady state probability of all states can be found. The explicit
expressions are included in Appendix-A

##  III Results and Discussions

Using the steady state probabilities of individual states, the velocity of
elongation ($V$) can be calculated as

$\centering V=\epsilon^{{}^{\prime}}P_{\mathrm{I_{\mathrm{C}}}}+\epsilon
P_{\mathrm{C}}.\@add@centering$ (6)

Here, $\epsilon^{{}^{\prime}}P_{\mathrm{I_{\mathrm{C}}}}$ and $\epsilon
P_{\mathrm{C}}$ represent the contributions to the elongation velocity by the
forward stepping to the next site after correct incorporation with stepping
rate $\epsilon$ and after incorrect incorporation with rate
$\epsilon^{{}^{\prime}}$ respectively. Plugging $P_{IC}$ and $P_{C}$ into Eqn.
(6), an explicit expression for $V$ can be obtained, which is given in
Appendix-A.

Further, the accuracy($A$) of DNA replication can be calculated as the ratio
of the correct flux and the total flux as

$\centering A=\dfrac{\epsilon P_{\mathrm{C}}}{\epsilon
P_{\mathrm{C}}+\epsilon^{{}^{\prime}}P_{\mathrm{I_{\mathrm{C}}}}}=\frac{1}{1+(\frac{\epsilon^{\prime}}{\epsilon})\frac{P_{IC}}{P_{C}}}.\@add@centering$
(7)

By substituting the expression for the steady state probabilities,
($P_{\mathrm{C}}$ and $P_{\mathrm{IC}}$) in the above expression, we can
obtain the accuracy $A$ of replication, which is given in the Appendix-A.

###  III.1 Purely polymerizing Scenario

To analyze the discrimination between correct and incorrect nucleotide and the
effect of errors in the velocity, we first consider the purely polymerizing
scenario, as observable in mutants without exonuclease activity. To that end,
we set $k_{ex}=k_{iex}=0$, i.e. the DNA polymerase does not make a transition
to the exonuclease state, neither from the correct binding state nor from the
incorrect binding state. For this case, the elongation velocity $V_{pp}$ can
be calculated as

$\begin{split}V_{pp}&=\dfrac{\varepsilon
k_{c}(\varepsilon^{{}^{\prime}}+k_{-ic})+\varepsilon^{{}^{\prime}}k_{ic}(\varepsilon+k_{-c})}{(\varepsilon+k_{-c})(\varepsilon^{{}^{\prime}}+k_{ic}+k_{-ic})+k_{c}(\varepsilon^{{}^{\prime}}+k_{-ic})}\end{split}$
(8)

Similarly, the accuracy $A_{pp}$ of replication can be calculated as

$\begin{split}A_{pp}&=\dfrac{\varepsilon
k_{c}(\varepsilon^{{}^{\prime}}+k_{-ic})}{\varepsilon\varepsilon^{{}^{\prime}}k_{c}+\varepsilon\varepsilon^{{}^{\prime}}k_{ic}+\varepsilon
k_{c}k_{-ic}+\varepsilon^{{}^{\prime}}k_{ic}k_{-c}}\end{split}$ (9)

Figure 2: Velocity $V_{\mathrm{PP}}$ and accuracy $A_{\mathrm{PP}}$ in the
purely polymerizing scenario, both (a,b) as a function of the binding rate of
incorrect nucleotides, $k_{ic}$ and (c,d) as function of the stepping rate
$\epsilon^{{}^{\prime}}$. Other parameters are varied as indicated in the
legends, the parameters $\epsilon=1$ and $k_{-c}=0.01$ are fixed.

Figure 2 (a) and (b) shows the plot of velocity $V_{pp}$ and the accuracy
$A_{pp}$ for a constant rate of correct nucleotide binding, varying the rate
of incorrect nucleotide binding ($k_{ic}$). It is observed that the
possibility of errors obviously decreases the accuracy whereas the influence
of such errors may increase or decrease the velocity of elongation. $V_{pp}$
may increase or decrease with $k_{ic}$, depending on the value of stepping
rate $\epsilon^{{}^{\prime}}$ after an erroneous incorporation. Similarly Fig.
2(c) and Fig. 2(d) shows $V_{pp}$ and $A_{pp}$ as a function of
$\epsilon^{{}^{\prime}}$ for different values of $k_{-ic}$. $V_{pp}$ has a
finite value at $\epsilon^{{}^{\prime}}=0$ and for any fixed value of
$k_{-ic}$. This nonzero value of $V_{pp}$ is due to the forward stepping of
the DNAP from the correct state to the next site with rate $\epsilon$. For
$\epsilon^{{}^{\prime}}=0$, the DNAP usually goes back to the active state for
a nonzero value of $k_{-ic}$ and eventually incorporation of a correct
nucleotide (and stepping) takes place. Thus for small $\epsilon^{\prime}$, the
typical fate of an incorrect nucleotide is to unbind, making the binding of an
incorrect nucleotide a dead-end branch off the main pathway and rapid
unbinding beneficial for high speed. By contrast, for large
$\epsilon^{\prime}$, incorrect nucleotides are incorporated and contribute to
the velocity of polymerization. Moreover, interestingly we observe that for a
critical value of $\epsilon^{{}^{\prime}}$, (i.e., at
$\epsilon^{{}^{\prime}}=\epsilon_{c}^{{}^{\prime}}$), at which the transition
between these two regimes occurs, the velocity is independent of $k_{-ic}$.
The accuracy (Fig. 2(d)), on the other hand, always increases when the
unbinding rate is increased, independent of the value of $\epsilon^{\prime}$.
Just like decreasing the binding rate of incorrect nucleotides, increasing the
unbinding rate $k_{-ic}$ always increases the accuracy, but may increase or
decrease the velocity.

The critical stepping rate $\epsilon_{c}^{{}^{\prime}}$ at which $V_{pp}$ is
independent of $k_{-ic}$ is exactly calculated as
$\epsilon_{c}^{{}^{\prime}}=\frac{\epsilon k_{c}}{\epsilon+k_{c}+k_{-c}}$ when
$k_{c}=k_{ic}$. From this expression one can notice that
$\epsilon_{c}^{{}^{\prime}}$ increases as a function of $k_{c}$ and for large
$k_{c}$ ($k_{c}\rightarrow\infty$), it approaches $\epsilon$, as plotted in
Fig. 3. This implies that $\epsilon_{c}^{{}^{\prime}}$ can be controlled by
fine tuning the $k_{c}$ values.

Figure 3: Critical stepping rate $\epsilon_{c}^{{}^{\prime}}$ as a function of
the binding rate $k_{c}$. The fixed parameters are $\epsilon=1$ and
$k_{-c}=0.01$.

The scenario considered so far, is the simplest exonuclease-deficient
scenario. Another exonuclease-deficient and therefore purely polymerizing
scenario is when transfer to the exonuclease site is possible, but with no
cleavage (i.e., with $k_{cl}=k_{cl}^{{}^{\prime}}=0$). Velocity and accuracy
in this case are plotted as a function of $k_{ic}$ together with the
corresponding results for the case without transfer to the exonuclease site
(i.e., for $k_{ex}=k_{iex}=0$) in Fig. 4. The velocity is generally reduced by
the transfer to the (inactive) exonuclease site, whereas the accuracy is seen
to be the same in both scenarios.

Figure 4: Sceanario with transfer to the exonuclease site, but without
exonuclease activity: Velocity $V$ and accuracy $A$ as a function of the
binding rate $k_{ic}$ of incorrect nucleotides for different values of the
stepping rate $\epsilon^{{}^{\prime}}$ and the exonuclase site transfer rates
$k_{ex}$ and $k_{iex}$. The other parameters are fixed: $k_{c}=k_{ic}=0.5$,
$\epsilon=1$, $k_{-c}=k_{-iex}=0.01$, and $k_{-ex}=0.001$.

###  III.2 Polymerization with exonuclease activity

Figure 5: Variety of behaviors in a scenario with exonuclease activity:
Velocity $V$ and accuracy $A$ as a function of the exonuclease transfer rate
$k_{ex}$ for two different values of $k_{-c}$ [in (a,b) and in (c,d),
respectively]. In all cases, there is no discrimination between correct and
incorrect nucleotides in the exonuclease pathway. The parameters are
$k_{c}=k_{ic}=0.5$, $\epsilon=1$,
$k_{\mathrm{cl}}=k_{\mathrm{-ex}}=k_{\mathrm{-iex}}=0.1$, and
$k_{\mathrm{cl}}^{{}^{\prime}}$ = $k_{cl}*\exp(3)$.
$k_{\mathrm{-ic}}=k_{\mathrm{-c}}*\exp(3)$. Note that both increase and
decrease of velocity and accuracy are seen and that the two are not generally
anticorrelated.  Figure 6: Three scenarios with discrimination between correct
and incorrect nucleotides in the exonuclease pathway: (a) Velocity $V$ and (b)
accuracy $A$ as a function of $k_{ex}$. The three cases depict discrimination
in the exonuclease transfer rate, the reverse transfer rate and the cleavage
rate, respectively. The fixed parameters are $k_{c}=k_{ic}=0.5$.

Including the exonuclease activity, we have exactly calculated the elongation
velocity, $V$ as well as the accuracy of replication, $A$ using Eq. (16) and
Eq. (19), respectively. We expect that the typical effect of a proofreading
mechanism is a reduction of the velocity and an increase in accuracy. However,
the following consideration shows that this is not generally the case: Without
discrimination between the correct and incorrect nucleotides in the
exonuclease transfer step ($k_{ex}=k_{iex}$), we observed that both velocity
as well as accuracy decrease with increasing $k_{ex}$ [Fig. 5(a) and 5(b)].
The decrease in accuracy can be understood as follows: Once a nucleotide,
correct or incorrect, is bound, it can either be incorporated irreversibly by
the forward step or be removed again by one of two pathways, unbinding or
exonuclease transfer and cleavage. If the cleavage rate is sufficiently high,
then the cleavage follows almost deterministically upon exonuclease transfer,
and the effective rate for the removal of the nucleotide is $k_{-c}+k_{ex}$
(or $k_{-ic}+k_{iex}$). If the first term discriminates between the correct
and the incorrect nucleotides, but the second does not, increasing the second
will inevitably reduce the accuracy rather than increasing it, as expected for
a proofreading pathway. Thus, additional discrimination in the exonuclease
pathway is required for proofreading.

We also note that the decrease of the velocity with increasing $k_{ex}$ is
also not universal. When there is no discrimination between the correct and
incorrect nucleotides in the exonuclease transfer step, for very small value
of $\epsilon^{{}^{\prime}}$, the velocity initially increases with $k_{ex}$
for very small values of $k_{-c}$, shows a maximum and then decreases with
$k_{ex}$ [Fig. 5(c)]. Whereas the accuracy increases with $k_{ex}$ [Fig.
5(d)]. The increase of velocity can be understood as follows: The presence of
incorrect nucleotide reduces the velocity due to the corresponding slow
stepping. Exonuclease can circumvent that slow stepping. However for high
exonuclease transfer rates, the velocity is decreased as exonuclease transfer
now also successfully compete with the rapid stepping after correct
incorporation. The presence of a maximum in the velocity is understood as
preventing slow stepping after incorrect nucleotide binding, but not rapid
stepping after correct binding, suggests one principle how the exonuclease
transfer rate should be set, namely between the two stepping rates (see
below).

These results shed some light on the question of a trade-off between velocity
and accuracy, which is generally expected for proofreading mechanisms and has
been reported in a number of studies [6, 27, 28]. Our results show that such
trade-off is also seen in our minimal model, provided the kinetic parameters
are in the range expected for DNA polymerases. The observation of a trade-off
is independent of the mechanisms of discrimination in the proofreading pathway
as discussed below. However, Figure 5 shows that speed-accuracy tradeoff is
not universal, consistent with a recent study [31], but rather dependent on
the kinetic model parameters. This is a typical feature of kinetic
proofreading mechanism and can be due to the discrimination of free energy
associated with different reactions.

Since additional discrimination between correct and incorrect nucleotides is
needed for the exonuclease activity to result in an increase in accuracy, we
next compared the three possibilities for discrimination in exonuclease
pathway: Using realistic parameters for the polymerization pathway, including
discrimination in unbinding ($k_{-ic}>k_{-c}$) and in stepping
($\epsilon^{\prime}<\epsilon$), we determined the velocity and the accuracy
for (i) $k_{iex}>k_{ex}$, (ii) $k_{-iex}<k_{-ex}$, and for (iii)
$k_{cl}^{\prime}>k_{cl}$. In all these three different cases, all other rates
of the exonuclease pathway are the same for correct and incorrect nucleotides,
and we take the rates that are different to differ by a factor of 10. The
latter factor is of the order of $e^{\frac{\Delta G}{k_{B}T}}$ with a mismatch
energy $\Delta G\sim 3k_{B}T$.

Fig. 6 shows the comparison between the three cases. In all cases, the
accuracy increases with increasing exonuclease transfer rate, while the
velocity decreases. However there are some marked differences between the
scenarios: Discrimination in the transfer rate or in the cleavage rate show an
almost identical decrease in the velocity and a very similar increase in
accuracy. However, the increase in accuracy is shifted towards higher values
of $k_{ex}$ for the case of discrimination in the transfer rate as compared to
the discrimination in the cleavage rate. Thus, for the same transfer rate (and
for the same decrease in velocity), the discrimination in transfer results in
a higher accuracy than discrimination in cleavage. Discrimination in the
reverse transfer rate, by comparison, shows an overall smaller increase in
accuracy as well as smaller decrease in velocity.

##  IV Conclusions

In this paper, we have studied a simple kinetic model of DNA replication and
have exactly calculated the accuracy and elongation velocity in the absence as
well as in the presence of exonuclease activity. In general, the fidelity of
DNA replication is based on the nucleotide selectivity of the DNA polymerase
during incorporation, its exonucleolytic proofreading activity and the
postreplicative DNA mismatch repair. The latter contribution is not included
in our model, in which an error remains uncorrected once the new DNA strand is
elongated and DNA polymerase has made a forward step. During the
exonucleolytic proofreading, an erroneously incorporated nucleotide is cleaved
off, hence the exonuclase activity improves the overall accuracy of
replication, but at the same time it also typically reduce the speed of
elongation by "resetting" the replication process to the beginning of the last
step. Similarly the escape of errors (via erroneous stepping) speeds up the
elongation mechanism but compromises the overall accuracy. The tradeoff
between speed and accuracy however is not universal, but depends on the
kinetic parameters, in agreement with the earlier results [31]. Our minimal
model provides a simple picture of the design of an accurate replication
system. Through the modulation of the model parameters, we can identify
several kinetic principles. Discrimination between the correct and incorrect
nucleotides arises both from the polymerization pathway as well as from the
proofreading pathway and is based on the kinetic competition in both the
cases. For both, slow forward stepping after incorporation of an incorrect
nucleotide (slower than for a correct nucleotide) is crucial for the accuracy,
giving time for unbinding and transfer to the exonuclease site, respectively.
However, additional discrimination in the exonuclease pathway is needed and
provided most efficiently by the transfer reaction. Thus the accuracy depends
on the coordinated action of the polymerase and exonuclease activity of DNA
polymerase.

##  V Acknowledgments

MS acknowledges the INSPIRE Faculty award (IFA 13 PH-66) by the Department of
Science and Technology, India and the Faculty Recharge Program (FRP-56055) of
UGC, India for the financial support.

##  VI Appendix

In this appendix, we give the explicit expressions for the steady-state
probabilities obtained as the solution to Eqs. (1)-(5):

$P_{\mathrm{C}}=k_{\mathrm{c}}(k_{\mathrm{-ex}}+k_{\mathrm{cl}})\\{k_{\mathrm{iex}}k_{\mathrm{cl}}^{{}^{\prime}}+(k_{\mathrm{-iex}}+k_{\mathrm{cl}}^{{}^{\prime}})(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}})\\}/T,$
(10)

$P_{\mathrm{I_{\mathrm{C}}}}=k_{\mathrm{ic}}(k_{\mathrm{-iex}}+k_{\mathrm{cl}}^{{}^{\prime}})\\{k_{\mathrm{ex}}k_{\mathrm{cl}}+(k_{\mathrm{-ex}}+k_{\mathrm{cl}})(\epsilon+k_{\mathrm{-c}})\\}/T,$
(11)

$P_{\mathrm{C_{\mathrm{EX}}}}=k_{\mathrm{ex}}k_{\mathrm{c}}\\{k_{\mathrm{iex}}k_{\mathrm{cl}}^{{}^{\prime}}+(k_{\mathrm{-iex}}+k_{\mathrm{-cl}}^{{}^{\prime}})(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}})\\}/T,$
(12)

$P_{\mathrm{I_{\mathrm{CEX}}}}=k_{\mathrm{iex}}k_{\mathrm{ic}}\\{k_{\mathrm{ex}}k_{\mathrm{cl}}+(k_{\mathrm{-ex}}+k_{\mathrm{cl}})(\epsilon+k_{\mathrm{-c}})\\}/T,$
(13)

and

$P_{\mathrm{A}}=1-P_{\mathrm{C_{\mathrm{EX}}}}-P_{\mathrm{I_{\mathrm{CEX}}}}-P_{\mathrm{I_{\mathrm{C}}}}-P_{\mathrm{C}}.$
(14)

In the above expressions, $T$ is given by

$\begin{split}T=k_{\mathrm{iex}}\\{k_{\mathrm{ex}}k_{\mathrm{cl}}^{{}^{\prime}}k_{\mathrm{c}}+k_{\mathrm{ex}}k_{\mathrm{cl}}(k_{\mathrm{ic}}+k_{\mathrm{cl}}^{{}^{\prime}})+(k_{\mathrm{-ex}}+k_{\mathrm{cl}})\\\
((k_{\mathrm{ic}}+k_{\mathrm{cl}}^{{}^{\prime}})(\epsilon+k_{\mathrm{-c}})+k_{\mathrm{cl}}^{{}^{\prime}}k_{\mathrm{c}})\\}+(k_{\mathrm{-iex}}+k_{\mathrm{cl}}^{{}^{\prime}})\\\
\\{k_{\mathrm{ex}}k_{\mathrm{ic}}k_{\mathrm{cl}}+k_{\mathrm{ex}}(k_{\mathrm{c}}+k_{\mathrm{cl}})(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}})+(k_{\mathrm{-ex}}+k_{\mathrm{cl}})\\\
(k_{\mathrm{c}}(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}})+(k_{\mathrm{ic}}+(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}}))(\epsilon+k_{\mathrm{-c}}))\\}.\end{split}$
(15)

We also give explicit expressions for the velocity and accuracy as obtained
from Eqs. (6) and (7): The velocity $V$ can be expressed as

$V=\dfrac{S}{U}$ (16)

with $S$ and $U$ are given by

$\begin{split}S=\epsilon^{{}^{\prime}}k_{\mathrm{ic}}(k_{\mathrm{cl}}^{{}^{\prime}}+k_{\mathrm{-iex}})\\{k_{\mathrm{cl}}k_{\mathrm{ex}}+(\epsilon+k_{\mathrm{-c}})(k_{\mathrm{-ex}}+k_{\mathrm{cl}})\\}\\\
+\epsilon
k_{\mathrm{c}}(k_{\mathrm{cl}}+k_{\mathrm{-ex}})\\{k_{\mathrm{cl}}^{{}^{\prime}}k_{\mathrm{iex}}+(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}})(k_{\mathrm{cl}}^{{}^{\prime}}+k_{\mathrm{-iex}})\\}\\\
\end{split}$ (17)

and

$\begin{split}U=(k_{\mathrm{cl}}^{{}^{\prime}}+k_{\mathrm{-iex}})\\{(k_{\mathrm{cl}}+k_{\mathrm{-ex}})((\epsilon+k_{\mathrm{-c}})(\epsilon^{{}^{\prime}}+k_{\mathrm{ic}}+k_{\mathrm{-ic}})\\\
+k_{\mathrm{c}}(\epsilon^{{}^{\prime}}+k_{\mathrm{-ic}}))+k_{\mathrm{ex}}(\epsilon^{{}^{\prime}}+k_{\mathrm{-c}})(k_{\mathrm{c}}+k_{\mathrm{cl}})+k_{\mathrm{ic}}k_{\mathrm{cl}}k_{\mathrm{ex}}\\}\\\
+k_{\mathrm{iex}}\\{(k_{\mathrm{cl}}+k_{\mathrm{-ex}})(k_{\mathrm{c}}k_{\mathrm{cl}}^{{}^{\prime}}+(\epsilon+k_{\mathrm{-c}})(k_{\mathrm{ic}}+k_{\mathrm{cl}}^{{}^{\prime}}))\\\
+k_{\mathrm{cl}}k_{\mathrm{ex}}(k_{\mathrm{ic}}+k_{\mathrm{cl}}^{{}^{\prime}})+k_{\mathrm{c}}k_{\mathrm{cl}}^{{}^{\prime}}k_{\mathrm{ex}}\\}.\end{split}$
(18)

The accuracy $A$ is found to be

$A=\dfrac{1}{1+\dfrac{\epsilon^{{}^{\prime}}}{\epsilon}N},$ (19)

where

$N=\dfrac{k_{ic}(k_{-iex}+k_{cl}^{{}^{\prime}})[k_{ex}k_{cl}+(k_{-ex}+k_{cl})(\epsilon+k_{-c})]}{k_{c}(k_{-ex}+k_{cl})[k_{iex}k_{cl}^{{}^{\prime}}+(k_{-iex}+k_{cl}^{{}^{\prime}})(\epsilon^{{}^{\prime}}+k_{-ic})]}.$
(20)

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