Abstract:
A method of processing sequencing data obtained with a polymer sequencing system identifies the most likely monomer sequence of a polymer, regardless of stochastic variations in recorded signals. Polymer sequencing data is recorded and two or more distinct series of pore blocking signals for a section of the polymer are recorded. A value is assigned to each series of pore blocking signals to obtain multiple trial sequences. The probability that each of the trial sequences could have resulting in all of trial sequences is calculated to determine a monomer sequence with the highest probability of resulting in all of the trial sequences, termed the first iteration sequence. The first iteration sequence is systematically altered to maximize the combined probability of the first iteration sequence leading to all the trial sequences in order to obtain a most likely sequence of monomers of the polymer.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application represents a continuation of U.S. patent application Ser. No. 12/614,782 entitled “Method for Deducing a Polymer Sequence from a Nominal Base-By-Base Measurement” filed Nov. 9, 2009, now U.S. Pat. No. 8,452,546, which claims the benefit of U.S. Provisional Patent Application Ser. No. 61/112,351 entitled “Data Processing Method for Deducing a Polymer Sequence from a Nominal Base-by-Base Measurement” filed Nov. 7, 2008. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with government support under Contract No. W911NF-08-C-0046 awarded by the Army Research Office and the Defense Advanced Research Projects Agency and Contract No. 2R44HG004466-02 awarded by the National Institute of Health, specifically, the National Human Genome Research Institute. The government has certain rights in the invention. 
    
    
     SEQUENCE LISTING 
     The instant application contains a Sequence Listing, which has been submitted as an ASCII text file via EFS-Web and is hereby incorporated by reference in its entirety. The ASCII text file, created on Mar. 8, 2013, is named SequenceListing.txt and is 26,639 bytes in size. 
     BACKGROUND OF THE INVENTION 
     Extensive amounts of research and money are being invested to develop a method to sequence DNA, (Human Genome Project) by recording the signal of each base as the polymer is passed in a base-by-base manner through a recording system. Such a system could offer a rapid and low cost alternative to present methods based on chemical reactions with probing analytes and as a result might usher a revolution in medicine. 
     Research in this area to date has focused on the question of developing a measurement system that can record a sufficient signal from each monomer in order to distinguish one monomer from another. In the case of DNA, the monomers are the well-known bases: adenine (A), cytosine (C), guanine (G), and thymine (T). It is necessary that the signals produced by each base be: a) different from that of the other bases, and b) different by an amount that is substantially larger than the internal noise of the measurement system. This aspect of sequencing is fundamentally limited by the specific property of the polymer being probed in order to differentiate the monomers and the signal to noise ratio (SNR) of the measurement device used to probe it. 
     A separate question is the order in which the monomers are measured. In order to know which monomer (or group of monomers) is being measured, it is necessary to localize the polymer to a precision comparable in length to the monomer itself. Controlling the polymer position and motion at such short length scales is challenging, in particular the polymer is subject to diffusive (Brownian) motion due to the impact of other molecules in solution. 
     One popular method to limit the polymer motion is to pass it through a nanopore, an approximately cylindrical cavity in a solid substrate with diameter equal to or a little larger than the polymer of interest. For such a nanopore, the polymer motion is effectively in one dimension (1-D) along the axis of the pore, but is still subject to stochastic variations in this 1-D motion due to Brownian effects. Specifically, Brownian motion results in a “random walk” such that the mean square displacement in a given time t is given by 2Dt for a polymer of diffusion constant D. This random motion is added to the imposed translocation motion, resulting in an inherent uncertainty in the number of bases that have passed through the measurement device. For example, for DNA confined within an alpha-hemolysin (αHL) protein pore at 15° C., the mean net 1-D motion due to diffusion alone in 100 microseconds (μs) is approximately 5 bases. Thus, in a notional example in which a given base is measured for 100 μs, the DNA would on average have moved a linear distance away from its desired position a total of 5 bases in either direction due to diffusion, resulting in, in this example, a segment of the DNA being re-measured or skipped. Such positional errors can occur no matter how sensitive the measurement system is that identifies each base. 
     Recent discoveries have shown that physical changes such as cooling the electrolyte and changing the viscosity of the electrolyte can reduce the diffusion constant of DNA in αHL by a considerable factor. However, even with such measures, methods proposed to sequence DNA by recording the signal of each base in a serial manner are still expected to have sequence order errors exceeding the current benchmark target of 99.99% accuracy. Accordingly, what is needed in order to develop a practical polymer sequencing system from such new approaches is a method to process data in order to reduce the effect of stochastic variations in the polymer position. 
     SUMMARY OF THE INVENTION 
     The method of the present invention utilizes a combination of data processing steps to limit the sequencing error produced by stochastic motion of a polymer in solution, and thereby improves the sequencing accuracy of the overall sequencing system. Initially, a polymer sequencing system, such as a nanopore sensing system, is utilized to record sequence data. The input data for the method of the present invention is an observation of the time series recording of the signal produced as a polymer passes through the nanopore of the sensing system. Two or more observations of each section of the polymer to be sequenced are made. The observations can be made by repeatedly measuring the same polymer molecule, by measuring multiple molecules of the same polymer, or by a combination of both methods. The first step in the data processing method of the present invention is to assign a value to each apparent monomer in each of the observations based upon foreknowledge of the signal amplitude produced by each monomer, and a physical model of the underlying process by which the polymer moves through the device. The outcome is a set of M (M≧2) trial sequences, one trial sequence for each observation. 
     The second step is to assume in turn that each particular trial sequence is true and calculate the total probability that this particular sequence could have resulted in all of the M observations. The total probability is calculated from the known statistics of the underlying stochastic processes that lead to the variations in polymer position. The trial sequence with the highest total probability of resulting in the complete set of observations is chosen as the first iteration. 
     In the preferred embodiment, a third step comprises systematically altering the first iteration sequence to maximize the combined probability of its leading to the M observations. In the simplest embodiment, it is assumed that all changes are local, involving only one or two adjacent monomers, and at each position a small set of likely changes is evaluated to see if any improves the combined probability, with the probabilities calculated as for the second step. If the combined probability improves, the change is kept and the process continues. This is done consecutively for each monomer position in the first iteration sequence, and then started again from the beginning and repeated until an entire sweep through the positions results in no further statistically relevant improvement. 
     The invention is not specific to the method used to identify an individual monomer and can in principle be utilized in combination with any method that seeks to sequence a polymer, or indeed any method that measures a property of polymer that is subject to stochastic variations in the order in which the monomers are measured, and in the duration of the event associated with each monomer. The invention also does not require that the polymer remain intact during the measurement process and applies to cases in which the observations are made on multiple individual molecules that are nominally the same. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts a polymer sequencing system for use with the method of the present invention; 
         FIG. 2A  is a graph depicting a model recording of pore current blockage signals over time for a 25-base ssDNA random sequence; 
         FIG. 2B  is a graph depicting a first simulated recording of pore current blockage signals over time for the 25-base ssDNA of  FIG. 2A ; 
         FIG. 2C  is a graph depicting a second simulated recording of pore current blockage signals over time for the 25-base ssDNA of  FIG. 2A ; 
         FIG. 2D  is a graph depicting a third simulated recording of pore current blockage signals over time for the 25-base ssDNA of  FIG. 2A ; 
         FIG. 3  is a graph of potential vs spatial coordination of a particle being sequenced; 
         FIG. 4  is a base identification model for use with the present invention; 
         FIG. 5  depicts a Sequence Hidden Markov Model; and 
         FIGS. 6A-6C  include illustrative sequencing information generated utilizing the method of the present invention identified as SEQ ID NOS: 1-114. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     With initial reference to  FIG. 1 , a polymer sequencing system or sensing system  1  is utilized in accordance with the present invention. In general, sensing system  1  includes a first fluid chamber or electrolyte bath  4  within which is provided a first solution or electrolyte  6 , and a second fluid chamber or sensing volume  8  provided with a second electrolyte  10 . Sensing volume  8  is separated from electrolyte bath  4  by a barrier structure  11 , which includes a nanopore or nano-scale orifice  17  that provides a fluid path connecting the first and second electrolytes  6  and  10 . In one preferred embodiment discussed herein, orifice  17  is in the form of a protein pore embedded in a lipid bilayer across an opening in barrier structure  11  in a manner known in the art. In general, sensing system  1  controls the translocation of a polymer indicated at  18  through orifice  17  utilizing a translocation means or means for controlling the velocity of a polymer through orifice  17  in the form of a power source  20 . In the embodiment shown, translocation power source  20  includes an AC bias source  22  and a DC bias source  23 . In addition, a current sensor  24  is provided to measure the AC current through orifice  17  produced by the AC bias source  22 . More specifically, current sensor  24  is adapted to differentiate monomers of a polymer on the basis of changes in the electrical current that flows through orifice  17 . In a manner known in the art, electrodes  28 ,  30 ,  32  and  34  are utilized in conjunction with current sensor  24  and power source  20 . A controller  52  may be utilized to control system  1 . Current signals detected or measured by current sensor  24  are processed in order to calculate a nominal monomer sequence of polymer  1 . For purposes of the present invention, the term “observation” is used to describe measurements of a region of a polymer taken over time utilizing a sensing system, such as sensing system  1 . The method of the present invention utilizes two or more repeat observations of the same region of one or more polymers. In general, sensing system  1  is provided for illustrations purposes only, and it should be understood that any equivalent means for detecting individual monomers of a polymer may be utilized in conjunction with the present method. 
     In accordance with the method of the present invention, sensing system  1  is in communication with means for deducing the sequence of a polymer from the nominal monomer sequences sensed by current sensor  24 , such as computer  50 . In a preferred embodiment, computer  50  includes software  54  configured to perform the method for deducing the sequence of a polymer of the present invention. Preferably, observations which are determined to have inadequate data quality (e.g., poor measurements) are excluded from the recorded observations processed by the method of the present invention. Computer  50  additionally includes an input device indicated at  56  for entering data, a display  58  for viewing information and a memory  60  for storing information. 
     The invention applies to polymers in general and to any method that seeks to sequence a polymer by measuring its monomers in a serial manner, but because of its technological significance and large body of existing experimental data, specific examples herein are discussed in terms of sequencing DNA. Further, because of its relative maturity and simplicity among serial-read methods, simulated data for the protein pore current blockade (PPCB) approach of sequencing DNA is utilized in order to illustrate the steps of the invention. However, it should be understood that the method of the present invention is not limited for use with DNA, nor the PPCB measurement method. 
     In the PPCB approach, a signal indicative of each base is obtained via the reduction in the ionic current passing through pore  17  when a base of polymer or DNA  18  is present in pore  17 .  FIGS. 2A-2D  illustrate the differences that may occur during separate sequencing sessions of the same DNA strand. More specifically,  FIG. 2A  illustrates a model recording of pore current blockage signals over time, without the effects of noise and diffusion, for a 25-base ssDNA random sequence of CCAGTTGACAAATGGCCCCTGTACA (SEQ ID NO: 115), wherein for the purposes of illustration, C is assigned a current blocking value of 3, A has a value of 2, G has a value of 0 and T has a value of 1.  FIGS. 2B-2D  depict three simulated PPCB signal recordings (or observation recordings) as a function of time for the translocating DNA sequence of  FIG. 2A , calculated for system parameters of −10° C., 10 kHz bandwidth, 0.33 pArms system noise and a 110 mV DC bias. A fit to each observation recording is shown by the solid line. An example of equivalent regions in  FIGS. 2B-2D  is given by the ellipses. 
     It is evident that, while very similar, the mathematically generated records in  FIGS. 2B-2D  contain differences in duration, despite being for the same input sequence of DNA. These differences arise because of the inherent uncertainties in the stochastic motion of the DNA through the pore. An immediate example is the factor of 2 variations in total translocation time for the three simulated datasets shown in  FIGS. 2B-2D . Looking more closely at the graph in  FIG. 2C  (reading from left to right), we see that the first transition from a 3 pA to 2 pA blockade current is missing as compared to  FIGS. 2B and 2D . A second example is the variation in duration of the first 1 pA level in  FIGS. 2B-2D . More specifically, in  FIG. 2D  the duration at this level is approximately 13 times longer than in  FIG. 2C , while in  FIG. 2B  it is approximately equal to the idealized dwell time in  FIG. 2A . 
     Although individual diffusion events are stochastic, the overall statistical distribution in the case of a diffusion process is well defined. For example, one dimensional diffusion along the axis of a nanopore can be described mathematically as the motion of a rigid particle in a periodic potential. The application of a voltage to pull the DNA through the nanopore corresponds to a tilt in the potential, as shown in  FIG. 3 . In the limit of large barrier height, the motion can be described as thermally activated hopping from one potential minimum to the next. 
     The height of the potential barrier (C0, C1) can be determined from the variation of the measured diffusion constant with temperature. For example, for single strand DNA (ssDNA) in the protein pore alpha hemolysin (αHL) the activation energy, E, for a simple series of adenosine (A bases) (at zero potential tilt)=1.8×10-19 Joules=108 kJ/mole. At room temperature (20° C.), the value of 1.8×10-19 Joules is 45 times kbT (where T is the temperature of the solution in Kelvins and kb is a constant, known as Boltzmann&#39;s constant), which shows that under all likely experimental conditions, ssDNA motion in αHL can be accurately described as thermally activated hopping between minima of the 1-D potential. 
     A particular impact of being in the thermal activated hopping regime is that the target molecule (e.g., DNA) to be sequenced hops at a time determined by the statistics of thermal activation. Thus, equivalent to the uncertainty in the direction and total number of hops in the position of the target polymer is uncertainty in the time when it moves. This means that whatever time interval is chosen between individual measurements, it is not possible to be certain the molecule has hopped to the next minimum (as desired), hopped to the previous minimum, stayed at the same minimum, or hopped two or more times, skipping one or more bases. To minimize the impact of a hop occurring before a measurement is completed, the measurement time should be minimized. However, it is generally the case that the shorter the measurement time the less accurate the measurement, so there is a practical limit to the data acquisition rate. 
     Activation over a potential barrier is a paradigm statistical process, the dynamics of which apply to chemical reactions such as enzymes that process DNA. Thus, while new approaches which utilize chemical synthesis or base cutting processes to control the order the bases are measured may seem quite different, in reality, the arrival and dwell time of the bases at the detection zone in these schemes are also subject to random fluctuations in a similar manner as discussed for PPCB-based sequencing. For example, consider the case where the DNA is cut into individual deoxy nucleotide monophosphates (dNMP) at the opening to the pore in a manner that allows the dNMPs to sequentially enter and block the pore. Some dNMPs will escape without blocking the pore and thus be skipped, some dNMP blocking events may happen too close together for both to be resolved (i.e., also skipped), and some may remain bound so long that they are counted as a repeat instance of the same base. In addition, it is not required that the stochastic nature of the disturbance of the base order be specifically thermally activated, and any process that introduces a variation can be addressed by the invention, provided that its statistical distribution can be characterized. 
     The starting point for the invention is that when the dynamics of the polymer motion are subject to stochastic processes, averaging multiple observations of the same polymer region will lead to meaningless intermediate data values that will only increase the error in identifying the monomers. Accordingly, the present invention utilizes an algorithm to quantize the data into an estimate of the monomer sequence as a first step, and only thereafter incorporates information from additional observations of the same sequence segment. 
     In the DNA example, if the measurement process has the amplitude of the signal caused by the identity of the single monomer at a sensitive site, one way to quantize the initial data into specific monomers is to apply a simple four state hidden Markov Model (HMM) with one state for each base type (shown in  FIG. 4 ). An HMM is a well known statistical model in which the system being modeled is assumed to be a Markov process with unobserved state. By applying the forward-Viterbi algorithm to each observation, the most likely series of bases can be extracted, along with the time duration the system spends at each base between state changes. The forward algorithm is known to those skilled in the art as an algorithm for computing the probability of a series of observed events. The output of this single base model is thus a sequence in which each base is different. 
     If the measurement process is such that multiple bases are measured at the same time and contribute to the recorded signal, then this first level model (HMM1) must have more states. For instance, if the sensitive region of the measurement encompasses two base positions, the model must have a state for each possible two base sequence, for a total of sixteen states (4 n  states where 4 is the number of base types and n is the estimated number of bases contributing to the measured signal). In this case, the signal produced is likely to have degeneracies (i.e., the base pair AT might produce the same signal as the base pair CG). These degeneracies may be resolved by considering the prior and following signals. For example, if the prior signal indicates the state GA and the following indicates TG, then the state AT is chosen over CG. The output of this two base model will also be a sequence in which each base is different. Conceptually, if the signal is produced by more than two bases, an n base model can be used with a state for each possible n base sequence. 
     A first estimate of the number of repeat instances of each base can be obtained from the duration the system spends in each state. Alternatively, the number of repeats can be estimated for each monomer at the end of Step 2 (discussed below) from the distribution of observation times for that monomer. 
     The first step of the algorithm of the present invention is equivalent to algorithms that are commonly utilized for extracting idealized ion channel currents from background noise. However, to improve upon these algorithms we note that, independent of the accuracy in distinguishing bases from one another, there will be errors in the order the bases arrive at the measurement system due to the influence of stochastic processes. Specifically, sensing systems, such as sensing system  1 , will occasionally jump to the next base (or further) within the measurement window allocated per base, or step back to a previous base. 
     To reduce the impact of the above statistical fluctuations, sequencing measurements in accordance with the present invention are performed a total of M times (M≧2) to produce a set of M observations, each observation consisting of a series of measurements. In one embodiment of the invention, the set of M observations can be obtained by measuring different polymers or regions of different polymers containing the same monomer sequence. In other words, multiple observations of the same region of one or more polymers are recorded using sensing system  1 . In another embodiment of the invention, the set of M observations can be obtained by measuring the same polymer or region of a polymer multiple times. One method of measuring the same polymer or region of the polymer involves reversing the direction of motion of the polymer through orifice  17  during sequencing. This reversal can be performed repeatedly to record multiple observations of the polymer during both the forward and backward motion of the polymer. 
     For each of the M observations recorded, a first level Hidden Markov Model (HMM1) is utilized to produce a likely sequence of bases or likely monomer sequence. We term each of these M sets of provisional serial base values a trial sequence. If an observation is of poor quality, as judged by the confidence level of the HMM1, or the signal to noise ratio of the measured data, or another metric, the observation can be discarded and additional data collected. We term the chosen set of M trial sequences a complete set of observations. In Step 2 of the method, we then use each of the M trial sequences to construct a Hidden Markov Model representing that sequence. This Sequence Hidden Markov Model (SHMM) is comprised of a base sequence X i , including for each base in the sequence a repeat number as illustrated in  FIG. 5 . The parameters needed to use the standard HMM algorithms are the transition probabilities T ij  defining the probabilities of a transition from state X i  to state X j , the start probabilities S i  defining the probabilities that the first measurement is of state X i , and the observation probabilities O ki  defining the probability that we observe the current I k  for duration D k  given that we are in the state X i  with base repeat R i . Also included is a finishing probability that the system exits from the state X i . The SHMM model differs from a strict HMM in that the transition probabilities and the observation probability depend not only on the present state, but also on the direction of the previous hop. 
     In our example of ssDNA being sequenced by the PPCB method, the probabilities are calculated from the well defined statistical processes of the activated hopping regime and measurements of the diffusion of known polymers. The measurements may indicate that the transition probabilities are dependent on the identity of the monomer being measured, the applied voltage, and on the direction of the motion of the polymer. 
     For example, the transition probability from state X i  to state X j  consists of two parts. First is the probability of a jump forward or backward, depending only on if i&gt;j or i&lt;j. If |i−j|=1, this is the complete transition probability from state X i  to X j . For |i−j|&gt;1, we must also include the probability that intervening states are skipped over in less than one measurement time. Similarly, the probability of a jump forward or backward depends not only on whether i&gt;j or i&lt;j, but also on the number of base repeats R i  in state X i , and, if R i &gt;1, on whether the last step was forward or backward. For example, for R i =2 we define P 2  as the probability of a forward jump given that the system still needs to make two forward jumps to get to the state X i+1 , and define P 1  as the probability of a forward jump given that the system needs to make only one more forward jump to get to the state X i+1 . From such considerations, we can then derive equations for the probability of a forward jump when the prior jump was forward or backward, respectively, and similar equations for the probability of a backward jump. This can be generalized to higher values of defining P k  as the probability of a forward jump when k forward jumps are needed to reach X i+1 , leading to a matrix of transition probabilities. 
     Each of the M SHMMs, each representing one of the M trial sequences, are run through a modified Viterbi algorithm against all of the M observations. This modified Viterbi algorithm is modified from the standard Viterbi algorithm to account for the dependence of the transition and observation probabilities on the direction of the previous step. The output of this modified Viterbi algorithm is the combined probability that the given SHMM produced all of the M observations. By quantifying the probabilities, we can identify the trial sequence, represented by its SHMM, with the highest combined probability as the first iteration F 1  in the search for the optimal sequence. 
     The third step is to systematically alter the F 1  sequence to maximize the combined probability of its leading to the M observations. In the simplest embodiment, we postulate that all changes are local, involving only one or two adjacent bases, and at each position we evaluate a small set of likely changes to see if any improves the combined probability. If the combined probability improves, we keep the change and move on. This is done for each position in order, until an entire sweep through the positions results in no further statistically significant improvement. Sequence variations are chosen from a set of statistically most likely changes. The changes used in the first embodiment along with examples of them, are summarized in Table 1 below. 
     Table 1. Local sequence changes to the F 1  used to generate an optimum sequence. 
     
       
         
               
               
               
               
             
           
               
                   
               
               
                 Change 
                 Description 
                 Permutations 
                 Example 
               
               
                   
               
             
             
               
                 Base  
                 Insert a nonmatching base 
                 2 
                 AG → ACG  
               
               
                 insertion 
                 before the current base 
                   
                 or ATG 
               
               
                 Double 
                 Duplicate current base  
                 1 
                 AGCT → 
               
               
                 insertion 
                 and following base 
                   
                 AGCGCT 
               
               
                 Base  
                 Change the current base  
                 1 or 2 
                 AGC → ATC 
               
               
                 change 
                 to a nonmatching base 
                   
                 AGA → ACA  
               
               
                   
                   
                   
                 or ATA 
               
               
                 Base  
                 Delete the current base 
                 1 or 0 
                 AGC → AC 
               
               
                 deletion 
                 (Avoiding repeat bases) 
                   
                 AGA → AA  
               
               
                   
                   
                   
                 not allowed 
               
               
                 Double  
                 Delete the current base  
                 1 or 0 
                 AGCT → AT 
               
               
                 deletion 
                 and the following base 
                   
                 AGCA → AA  
               
               
                   
                 (Avoiding repeat bases) 
                   
                 not allowed 
               
               
                 Change  
                 Change the current  
                 Undetermined 
                 A3 → A4 
               
               
                 base 
                 base repeat count 
                   
                   
               
               
                 repeat 
               
               
                   
               
             
          
         
       
     
     As an example, the specific embodiment of the invention described above was run on computer generated data sets of the type shown in  FIG. 2 . A total of 57 random 25-base sequences were input into the model and 25 possible time records were generated for each of the 57 random input sequence.  FIGS. 6A-6C  illustrate the 57 trial sequence pairs identified as SEQ ID NOS: 1-114, where the top sequence of each pair shown in  FIGS. 6A-6C  is the randomly generated “true” sequence, and the lower sequence of each pair shown in  FIGS. 6A-6C  is the optimum sequence generated by the invention. Characters underlined and in bold indicate the location and nature of an error in the calculated optimum fit. 
     Although described with reference to a preferred embodiment of the invention, it should be understood that various changes and/or modifications can be made to the invention without departing from the spirit thereof. For example, although an AC current blocking sensor is utilized in the example of a possible means for detecting polymer pore blocking signals, it should be understood that a DC current sensing system or other known monomer detecting system can be utilized with the present invention. In general, the invention is only intended to be limited by the scope of the following claims.