A mathematical model for a biphasic DNA amplification reaction

Abstract

Isothermal DNA amplification reactions are a prevalent tool with many applications, ranging from analyte detection to DNA circuits. Exponential amplification reaction (EXPAR) is a popular isothermal DNA amplification method that exponentially amplifies short DNA oligonucleotides. A recent modification of this technique using an energetically stable looped template with palindromic binding regions demonstrated unexpected biphasic amplification and much higher DNA yield than EXPAR. This ultrasensitive DNA amplification reaction (UDAR) shows high-gain, switch-like DNA output from low concentrations of DNA input. Here we present the first mathematical model of UDAR based on four reaction mechanisms and show the model can reproduce the experimentally observed biphasic behaviour. Furthermore, we show that three of these mechanisms are necessary to reproduce biphasic experimental results. The reaction mechanisms are (i) positively cooperative multistep binding spurred by two trigger binding sites on the template; (ii) gradual template deactivation; (iii) recycling of deactivated templates into active templates; and (iv) polymerase sequestration. These mechanisms can potentially illuminate the behaviour of EXPAR as well as other nucleic acid amplification reactions.

1. Introduction

DNA amplification technologies are a cornerstone of molecular diagnostics. While polymerase chain reaction (PCR) provides a gold standard for accuracy and sensitivity of nucleic acid detection, it requires complex laboratory equipment and trained personnel. Isothermal DNA amplification reactions that require less energy, time and equipment than PCR [1] have become an increasingly popular alternative for molecular detection. These reactions can even be run without electricity [2], which is particularly attractive for point of care and limited resource molecular diagnostics. Identification of specific DNA or RNA sequences in these situations could inform medical professionals and potentially curb health crises. Exponential amplification reaction (EXPAR) is a popular isothermal DNA amplification method that exponentially replicates short oligonucleotide triggers [3]. The central elements of EXPAR design are (1) a DNA trigger oligonucleotide that is either the target molecule or is produced by a target molecule, (2) a single-stranded DNA template and (3) polymerase and nickase enzymes. The trigger binds to a complementary region on the single-stranded DNA template and then is extended by the polymerase enzyme through a restriction site and a second identical complementary region. Extending over the second complementary region creates a copy of the original trigger which then is released from the template by a nicking endonuclease that recognizes the restriction site. Repeated extension and nicking of these complexes form a core amplification loop and lead to a rapid increase in trigger concentration. As the short oligonucleotide triggers can be created by proteins [4,5], microRNA [6], DNA [7,8] and RNA [9], amplification reactions of this type are a popular tool for molecular recognition schemes. These types of reactions also are used in DNA circuits [10] that show promise in synthetic biology.

Recently, we constructed an isothermal DNA amplifying template where the complementary binding regions contain a palindromic sequence. This construction produces intriguing biphasic behaviour [11] (figure 1). The pair of palindromic sequences cause the templates to fold into an energetically favourable hairpin structure which remains closed in the absence of trigger. To ensure the template opens in the presence of the appropriate trigger, the binding regions are enhanced with toeholds [13]. The triggers bind to the toeholds to initiate strand displacement, open the hairpin and begin the amplification process. The initial behaviour of these templates shows a quick amplification burst followed by a slow rising plateau similar to EXPAR. Unlike EXPAR, however, the plateau is interrupted by a secondary amplification phase where total DNA produced reaches 10–100 times the levels of standard EXPAR. Since this second phase greatly increases output signal, it is important to understand the factors that cause amplification to resume in this biphasic reaction. The goal of this work is to first mathematically model key reaction mechanisms suggested in [11] that may explain each phase of this reaction and then to simulate the mechanisms with reasonable approximations for each kinetic component. We compare this model to experimental data produced from an ultrasensitive DNA amplification reaction (UDAR) template and show that without the key reaction mechanisms, the model deviates from the data. Our ultimate goal is to predict and control the timing, duration and trigger concentration at the plateaus based on known DNA association and dissociation kinetics. To achieve this, we constructed a differential equation model based on four proposed reaction mechanisms.

• (i)Multistep binding. A trigger bound at the 3^′ end of the template can open the template and create an amplifiable complex. However, if a second DNA trigger does not bind quickly to the palindromic binding region formerly concealed in the loop, it is energetically favourable for the template to reform the stable hairpin form and knock off the first binding trigger than create an amplifiable complex. For example, the melting temperature of the looped template structure used in this study is 11.3°C higher than that of a single trigger associated with the template and the template favours the closed configuration. Thus, the forward reaction is extremely unlikely when only a single DNA trigger is bound. Because of this, we simplify the reaction for modelling purposes by assuming that a second trigger must bind to the template simultaneously to prevent the template from returning to the closed form. A second trigger molecule can only bind after a first trigger has opened the loop, so we consider this a positively cooperative multistep binding event. The trigger will more likely open the loop at the 3^′ toehold as the external toehold has association kinetics 10–100 times higher than the internal toehold located inside the loop [14].
• (ii)Template deactivation. During the amplification process, an infrequent polymerase error leaves a complementary top strand bound to the template in a complex that cannot be nicked. This reaction mechanism was posited by Rondelez et al. [15], but the incidence and prevalence of this process have yet to be investigated. We assume this mechanism causes the deactivation of amplifier complexes which, in turn, dramatically reduces trigger production and often causes the appearance of the first plateau.
• (iii)Template recycling. We hypothesize that the secondary phase of the trigger amplification is due to an accumulation of trigger replicates that can associate with one of four toehold enhanced palindromic binding sites on the deactivated amplifier. These triggers can aid in dislodging the complementary top strand that deactivated the amplifier and rescued the original template, which can then reform the original looped structure and become a functional amplifier again. Because the top strand is complementary, it also has two palindromic binding sites and an affinity for a closed loop form. Because of the energetic favourability of binding with the excess replicated trigger or creating a closed loop form, it is less likely for the complementary strands of a UDAR template to reassociate than for the complementary strands of a standard EXPAR template.
• (iv)Polymerase sequestration. Finally, the palindromic trigger can bind to itself and be extended by polymerase to produce an inactive dimer. This reaction was shown to occur when complementary palindromic regions are included in EXPAR templates [11,16] and was hypothesized to slow the reaction due to limiting the concentration of free trigger molecules. This model incorporates this effect and the subsequent competition for DNA polymerase. This competition limits the amplification reaction rate and results in a slow down of DNA production. The polymerase concentration used in UDAR and EXPAR experiments is typically approximately 5 nanomolar and will become limiting as the reaction proceeds to produce 10–100 micromolar of product.

Figure 1.

Biphasic DNA amplification output: representative outputs of the biphasic DNA amplification reaction UDAR. Type I and II templates can be distinguished by the free energy of DNA association, which is described in detail in [11]. Both types of templates exhibit two growth phases, but in type I templates, the first growth phase is slower than the second growth phase and the first plateau is relatively long. Type II templates show a rapid first phase of growth and slower second phase growth; the first plateau is relatively short. Type II experimental data are modelled in this study. The EXPAR template was reproduced from [12]. The fluorescent signal is produced by the DNA dye SYBR Green II intercalating into the reaction products.

Using reasonable parameter estimates based on values reported in the literature, we found that this model explains all key characteristics of the biphasic reaction described in [11]. Our model reproduces two distinct DNA amplification periods and two periods of stalled growth as observed in UDAR trials. We show that models without the last three mechanisms described above fail to reproduce biphasic behaviour. Our results support the hypothesis that the template deactivation hinders EXPAR amplification, which was originally ascribed to deactivation of the nickase [3].

Mathematical models of DNA amplification reactions are not common in the literature, presenting a significant challenge for researchers working on a broad range of applications. This work is relevant for general kinetic modelling of DNA amplification reactions used in DNA circuits [10], as well as quantitative molecular recognition [17].

2. Model description

The main chemical reactions we assume to occur during UDAR trials are presented in figure 2. The variables in our model represent the molar concentrations of different types of oligonucleotides measured in micromolar (μM). We refer to a molecule type by the same name as its concentration, i.e. the concentration of trigger oligonucleotide is denoted by [Y] and we will refer to an oligonucleotide of this type as trigger Y. We now describe the assumptions on which our model is based. All equations assume mass action kinetics, except the processes of template opening and template recycling. The individual reaction components are described below.

• —Let Y be a DNA trigger, which is the short DNA oligonucleotide that triggers its own amplification. Let T be a UDAR template consisting of two toehold enhanced palindromic binding sites separated by a restriction site. Because UDAR templates have a stronger affinity for the closed loop configuration than the open configuration with a single trigger bound, we simplify the equations by assuming single trigger bound template cannot extend and that two Ys must cooperatively bind to the template to initiate amplification. Let Z be this template with two bound triggers. The simplified multistep binding reaction creates Z forms at a rate of k₊ and the rate of the inverse reaction is k₋. The rates k₊ and k₋ that characterize this multistep process are estimated as described in the electronic supplementary material.
• —Let W be a complete amplifier, the double-stranded complex formed when polymerase extends a Z complex from the 3^′ to the 5^′ end of the template. As polymerase extends the Y bound to the template’s 3^′ end, it displaces the Y bound to the 5^′ end. The rate of extension is proportional to the availability of polymerase with rate constant k₁.
• —Let V be a nicked amplifier, a partially double-stranded complex formed when nickase releases the replicated Y from W. The rate of nicking is k₂ and V are extended into W proportional to the availability of polymerase at a rate k₃. We assume the process of extending V deactivates complete amplifiers at a rate k₄ and denote these deactivated amplifiers by W*.
• —Because Y can weakly dimerize in sufficient concentrations, the abundance of Y replicated in UDAR trials will begin to sequester itself into inactive dimers, D [11,16]. Y molecules associate at their palindromic regions and available polymerase extends through the toeholds, creating strongly associated dimers. The rate of formation of D is proportional to the amount of available polymerase and trigger with rate constant k₆. The initial trigger dimerization event has association constant κ_A. For the experimental data used in this study, the palindromic regions of the T and Y are 6 nucleotides long and Y weakly dimerizes at the reaction temperature.
• —W and W* consist of two strands; the original template T and a complementary top strand, J. J has a Y embedded in both ends with a restriction site between; see figure 2. We assume the Y molecules embedded in J or their complementary binding sites in T may be exposed by fraying ends which can lead to separation of complete and deactivated amplifiers, W and W*, into single strands T and J. This template recycling occurs at a rate k₅. Because there are four sites where Y can initiate strand separation, we model the recycling reaction as positively cooperative with a Hill function using Y as the ligand, Hill coefficient 4 and half-saturation constant κ_R. We note that D may also aid in template recycling by separating into individual strands that can bind to both the T and J strands of an amplifier at the same binding sites as Y. However, the strongly associated D produced by type II templates such as the one modelled in this study is energetically stable at the reaction temperature and therefore sequestered in the double-stranded form. Accordingly, in this model, we only consider Y-mediated template recycling. Both T and J can either form a stable looped structure or immediately bind with two more Y molecules. For T, this restarts a core amplification loop. For J, this creates an inactive byproduct G, which we describe next.
• —Because two triggers are embedded in the top strand, excess Y can weakly bind to J at either of these locations with the same affinity with which J forms the closed loop. Y bound to J in this manner can be extended by polymerase at a rate k₆ to form G. Because a Y bound on the 3^′ end will extend through the restriction site and release any Y that may have previously bound on the 5^′ end, we assume the process of forming G only removes one Y molecule from the pool of free Y molecules. When a Y bound to the 5^′ end is removed, it may have been extended by polymerase to form a strand identical to a single strand of the inactive dimer D. This strand then has an affinity for both free Y and strands identical to itself, either of which will then form D. Since Y bound on the 5^′ end of J is not extendable to a double-stranded G, we assume for simplicity that Y only binds to J at the 3^′ end.
• —
  Because the polymerase becomes occupied during several reactions, the concentration of free polymerase, P, is computed as

  $$[P]:=\frac{P_0}{1+(([Z]+[V])/{\kappa }_{P_1})+({\kappa }_A([J][Y]+{[Y]}^2)/{\kappa }_{P_2})},$$ 2.1

  where P₀ is the initial concentration of polymerase, ${\kappa }_{P_1}$ is the dissociation constant for oligonucleotides longer than 10 nt, ${\kappa }_{P_2}$ is the dissociation constant for oligonucleotides shorter than 10 nt and κ_A is the association constant for the dimerization event described above.

Figure 2.

Full model. (Online version in colour.)

The resulting set of equations to measure concentrations of each DNA form is

$$\begin{array}{rl}[\dot{Y}]& =-2k_{+}[T]{[Y]}^2+2k_{-}[Z]+k_1[Z][P]+k_2[W]-k_6[P][Y](2[Y]+[J])\\ [\dot{T}]& =-k_{+}[T]{[Y]}^2+k_{-}[Z]+k_5\left(\frac{{[Y]}^n}{{\kappa }_R^n+{[Y]}^n}\right)([W]+[W^{\ast }])\\ [\dot{Z}]& =k_{+}T{[Y]}^2-k_{-}[Z]-k_1[Z][P]\\ [\dot{W}]& =k_1[Z][P]-k_2[W]+k_3[V][P]-k_4[W]-k_5\left(\frac{{[Y]}^n}{{\kappa }_R^n+{[Y]}^n}\right)[W]\\ [\dot{V}]& =k_2[W]-k_3[V][P]\\ [{\dot{W}}^{\ast }]& =k_4[W]-k_5\left(\frac{{[Y]}^n}{{\kappa }_R^n+{[Y]}^n}\right)[W^{\ast }]\\ [\dot{J}]& =k_5\left(\frac{{[Y]}^n}{{\kappa }_R^n+{[Y]}^n}\right)([W]+[W^{\ast }])-k_6[J][Y][P]\\ [\dot{G}]& =k_6[J][Y][P]\\ \mathrm{and}[\dot{D}]& =k_6{[Y]}^2[P],\end{array}\}$$ 2.2

where the Newton dot derivative notation indicates a derivative with respect to time. Furthermore, we compute free polymerase [P] using (2.1) and impose the constraint

$$[T]:=T(0)-[Z]-[V]-[W]-[W^{\ast }].$$ 2.3

The initial conditions are

$$\begin{array}{rl}& (Y(0),T(0),P(0),Z(0),W(0),V(0),W^{\ast }(0),J(0),G(0),D(0))\\ & =(Y_0,T_0,P_0,0,0,0,0,0,0,0).\end{array}$$

2.1. Parameter ranges

Our model contains 12 parameters:

$$k_{+},k_{-},k_1,k_2,k_3,k_4,k_5,k_6,{\kappa }_R,{\kappa }_{P_1},{\kappa }_{P_2},{\kappa }_A.$$

Most of these parameters have known ranges.

• —k_+/− (binding two Y to one T). The rate of binding for a 10 nucleotide complex at a range of temperatures is reported in [18]. Using these correlations, we calculated the rate of a single trigger association with a template as 37.8 μM⁻¹ s⁻¹ and trigger dissociation as 7.9 s⁻¹ for a 10 nucleotide trigger. The rate of binding of two triggers to a looped template is not specifically available in the literature. To estimate the on and off rates of this association, we used Matlab to fit estimates of the first and second trigger association (bimolecular rate constants) to dual trigger association (trimolecular rate constants). Details of this association are given in the electronic supplementary material. Because k_+/− primarily affects the initiation of the first phase of trigger expansion, these parameters shift the time of the initial rise. The final parameters were chosen to fit the reaction initiation observed in experimental data for the template studied in this work while keeping reasonable values for the DNA association kinetics. The free energy of the trigger association to the looped template, given by the difference in free energy between the 3^′ toehold association with the trigger and the loop secondary structure, was one parameter shown in [11] to correlate with the inflection point of the first phase. This correlation is consistent across the biphasic amplification reactions originally reported. It is likely that k_+/− will shift the first phase initiation for other biphasic amplification reactions.
• —k₁ (extension of Z to W). The previously reported parameter for a polymerization event of a transient complex for traditional EXPAR is 2 s⁻¹ [3]. We adjusted this parameter to reflect the addition of available polymerase in the equation using the initial value of polymerase, P₀ = 0.005 μM, making k₁ approximately 2/P₀ = 400 s⁻¹ μM⁻¹.
• —k₂ (nicking of W). The original EXPAR model calculated nicking and extension of an amplifying complex (W) as 0.4 s⁻¹ [3]. We chose a similar rate to represent the nicking step.
• —k₃ (extension of V to W). For the template observed in this study, a nicked amplifier V has 12 unbound nucleotides while a double bound template Z has 22. Because extending V requires filling in about half as many nucleotides as filling in Z, we chose this parameter to be two times higher than extension of Z to W.
• —k₄ (template deactivation). Analysis of UDAR trials with the template observed in this study showed that an initial concentration of 0.1 μM of template produced 5 μM of a trigger before pausing at the first plateau. We therefore assume that templates become deactivated after approximately 50 extension events, corresponding to a deactivation rate approximately 50 times lower than the extension rate k₃ × P₀. Our chosen parameter is in this range.
• —k₅ and κ_R (recycling). κ_R is the concentration of trigger at which recycling operates at half of its maximum rate. We chose this to be between the concentration of the first plateau (approximately 5 μM) and the second plateau (approximately 50 μM). k₅ is a free parameter describing the recycling rate. It was chosen to fit the data as template recycling is a completely unexplored phenomenon.
• —k₆ (inactive dimer formation). This parameter has an upper bound calculated by employing a quasi-steady state assumption: association of the triggers to form a dimer is assumed to equilibrate quickly compared to the association of dimers with the polymerase. Details of this calculation can be found in the electronic supplementary material.
• —${\kappa }_{P_{1,2}}$ (polymerase dissociation constants). The affinity of DNA polymerase to DNA substrate is highly variable and can change with parameters such as DNA sequence. For example, we found measurements of polymerase dissociation constant that ranged from 5 to 200 nM. The trials observed in this study use Bst, a family A polymerase. Complete extension of partially double-stranded complexes Z and V, which require more than 10 base pairs to be filled in, is modelled with a dissociation constant ${\kappa }_{P_1}=15\hspace{0.17em}\text{nM}$. This is similar to the K_d = 5 nM of Pol I [19], a different family A polymerase. Another family A polymerase showed decreased initiation kinetics when primers were less than 10 bp [20]. These primers likely have a higher dissociation constant than longer priming regions. For this reason, we chose a value of the order of the highest dissociation constant found in the literature, 225 nM for ${\kappa }_{P_2}$ [21]. This order is appropriate for binding the short primer strands (6 nt) created during dimerization of Y.
• —κ_A (association constant of palindrome–palindrome binding). The association constant for two palindrome sites, such as when a free trigger binds to itself or a palindrome in an unbound recycled top strand, was calculated using the DINAmelt web server [22].

2.2. Measuring DNA

The measured reaction output is created by the fluorescent intercalating dye SYBR Green II, which produces a signal assumed to be approximately proportional to the length of DNA. To compare the model simulations to experimental results, simulated concentrations were converted to the number of nucleotides incorporated into the DNA reaction products. The product of the reaction was shown to be primarily free trigger (Y) with a growing number of inactive dimers (D). The approximate concentration of reaction products was determined using calibrated SYBR Green II fluorescence of 12 nucleotide triggers. Experimental data and model output are both reported as the equivalent μM of incorporated nucleotides. All oligonucleotides in our model are comprised of three fundamental components: the toehold, the palindrome binding site and the restriction site; see figure 3. Each strand length is expressed in nucleotides and can differ between experiments. We denote the length of toehold by $\mathcal{T}$, the length of the palindrome site by $\mathcal{P}$ and the length of the restriction site by $\mathcal{R}$. The length of each oligonucleotide can be expressed in terms of the three basic lengths, figure 3 (right). We note that G can take on two forms, G₁ or G₂, based on which trigger binds first. However, consistent with the assumptions described previously, we estimate total DNA in G form by the length of G₂. The total DNA in terms of incorporated nucleotides is

$$\begin{array}{rl}\mathfrak{T}=& \hspace{0.17em}(\mathcal{T}+\mathcal{P})[Y]+(2\mathcal{T}+2\mathcal{P}+\mathcal{R})([T]+[J])+(4\mathcal{T}+4\mathcal{P}+\mathcal{R})[Z]\\ & +(3\mathcal{T}+3\mathcal{P}+2\mathcal{R})[V]+(4\mathcal{T}+4\mathcal{P}+2\mathcal{R})([W]+[W^{\ast }])\\ & +(4\mathcal{T}+2\mathcal{P})[D]+(6\mathcal{T}+4\mathcal{P}+2\mathcal{R})[G].\end{array}$$

Figure 3.

Length and structure of various oligonucleotides in terms of three basic sequences. (Online version in colour.)

3. Results

Our model, constructed from proposed reaction mechanisms (i)–(iv) with parameters within biologically plausible ranges, is able to closely match experimental amplification from UDAR trials first reported in [11]; see figure 4. The reaction has five distinct phases.

• (I)Initiation, a period in which total DNA stays at near initial levels.
• (II)Primary amplification, a brief period where total DNA rapidly increases.
• (III)Plateau, a period where the growth rate of total DNA significantly decreases.
• (IV)Secondary amplification, a second amplification period where total DNA increases 10 fold.
• (V)Slow final growth, a final period where the total DNA growth rate decelerates.

Figure 4.

Simulation fit to data. (a) Fit of experimental data to total DNA predicted by the full model; parameters are within range of best estimates for the experimental conditions. Individual concentrations of each reaction product are also shown. (b) Detail (note the scale on the vertical axis) of non-monotonic behaviour of concentrations that drives the biphasic behaviour. (Online version in colour.)

The proposed reaction mechanisms successfully reproduce these features. We justify the connections between the reaction mechanisms and the above phases as follows.

• (I)The initial concentration of T is much larger than the initial concentration of Y. Mechanism (i) requires multistep binding to initiate amplification, so the reaction is limited by an insufficient amount of Y to bind two molecules to a T at once. Assuming multistep binding does occur infrequently during initiation, a small concentration of W will be produced. These complete amplifiers begin core amplification loops: W is nicked to form V, releasing a replicated Y, then being extended back into W to close the loop. However until a sufficient concentration of Y is replicated, this phase appears static.
• (II)The transition from initiation to primary amplification exhibits behaviour similar to ultrasensitive switches as reported by [23]; see figure 4b. When Y approaches the critical concentration where it is no longer limiting for the multistep binding process, T will have sufficient trigger available to form stable Z complexes. These then rapidly transform into W complexes and abundantly replicate Y. This drives the jump in amplification rate that characterizes the primary amplification phase.
• (III)As posited in mechanism (ii), W amplifiers are deactivated by polymerase errors during extension. As the W concentration rises, this deactivation becomes more pronounced. The accumulation of W* and decline in W, observed in figure 4b, reduces the availability of operational core amplification loops, slows the production of Y and initiates the first plateau. During the plateau, most W amplifiers are deactivated and therefore this phase is characterized by extremely subtle amplification. Mechanism (iii), however, postulates that Y can split deactivated W* complexes into separate hairpin shaped J and T strands. Because the W and W* structures are energetically favourable compared to Y association with the already bound palindromic regions of W, recycling is delayed until the Y is concentrated enough to displace J. As there is continued slow growth in Y during the plateau phase, Y will eventually attain the critical concentration required to start recycling W and W*.
• (IV)When this second critical concentration of Y is attained, we again observe switch-like behaviour as recycling becomes energetically favoured. The critical concentration of Y that triggers recycling is much higher than the critical concentration of Y needed for template opening, so recycled templates almost immediately form W complexes. Furthermore, this concentration of Y is sufficient to recycle W*. This means the amplification process at this stage follows a secondary amplification loop: two Ys immediately bind to the newly free T and extend to form W. W will then complete approximately 50 core amplification loops before being deactivated and recycled (or directly recycled without deactivation). This secondary loop drives the secondary amplification phase.
• (V)The final amplification deceleration is caused by mechanism (iv), the polymerase becoming saturated by an increased demand for polymerase late in the reaction. The concentration of Y and J becomes so abundant that weak trigger dimerization becomes a dominant reaction. When Y binds to another Y or a J, the polymerase extends to either synthesize the complementary toeholds (creating D) or synthesize the entire length of J (creating G). As the concentration of Y is much higher than the concentration of J, D is produced more rapidly than G. These reactions compete with core and secondary amplification loops for the polymerase. This competition for polymerase sequesters its availability and slows all reactions, as observed towards the end of the experimental time frame. The lack of available polymerase leads to the accumulation of V in the core amplification loop, accumulation of Z in the secondary amplification loop and a corresponding decrease in the concentration of W. Since W produces Y, this decreases the production of Y.

3.1. Partial models

To show the proposed mechanisms are essential to reconstruct all five phases of the reaction, we present the simulated results for a series of simplified models that satisfy some, but not all, of these hypotheses:

• (i)Model 1: No deactivation or recycling of amplifiers, omitting mechanisms (ii)–(iii).
• (ii)Model 2: Deactivation but no recycling of amplifiers, omitting mechanism (iii).
• (iii)Model 3: Polymerase is not limiting, omitting mechanism (iv).

In each partial model, we use the same parameters as the full model but set parameters corresponding to omitted mechanisms equal to zero. We numerically simulate the resulting model from the same experimental initial conditions as the full model.

3.2. Model 1: no deactivation or recycling of amplifiers

In this model, we use only mechanisms (i) and (iv). We represent this in the model by setting

$$k_4=k_5=0.$$

All other parameters are the same as in table 1. In this simplified model,

$$[W^{\ast }]\equiv 0,[J]\equiv 0,[G]\equiv 0.$$

The numerical simulation, shown in figure 5, does not reproduce the five phases that characterize the UDAR trials. It reproduces (I) initiation, (II) primary amplification and (V) slow final growth, but does not exhibit the initial plateau or secondary amplification phases. Deactivation reduces the number of active core amplification loops, so the omission of deactivation reduces the time required to achieve the critical concentration of Y needed to trigger the primary growth. This model does not reproduce the plateau because the only limitation on the growth of the DNA is sequestration of polymerase by the formation of inactive dimers D.

Table 1.

Model parameters.

parameter	range	simulation value	citation
k+	$\mathcal{O}({10}^2-{10}^3)\hspace{0.17em}\mu {\text{M}}^{-2}\hspace{0.17em}{\text{s}}^{-1}$	2000 μM−2 s−1	[18], electronic supplementary material
k−	$\mathcal{O}(1-10)\hspace{0.17em}{\text{s}}^{-1}$	45 s−1	[18], electronic supplementary material
k1	400 μM−1 s−1	400 μM−1 s−1	[3]
k2	0.4 s−1	0.625 s−1	[3]
k3	$\mathcal{O}(k_1)\hspace{0.17em}{\mu \mathrm{M}}^{-1}{\mathrm{s}}^{-1}$	$800\hspace{0.17em}{\mu \mathrm{M}}^{-1}{\mathrm{s}}^{-1}$	—
k4	$\mathcal{O}\left(\frac{k_1{P}_0}{50}\right)=\mathcal{O}(0.04)\hspace{0.17em}{\text{s}}^{-1}$	0.013 s−1	[3,11]
k5	—	0.1 s−1	—
κR	—	14 μM	—
k6	less than k1κA = 3 μM−2 s−1	0.01 μM−2 s−1	[22], electronic supplementary material
κA	7.5 × 10−3 μM−1	0.0075 μM−1	[22]
${\kappa }_{P_1},{\kappa }_{P_2}$	0.001–0.2 μM	0.015, 0.225 μM	[19–21]

Figure 5.

Partial model 1 (k₄ = k₅ = 0). (a) Total DNA and individual reaction products, shown with experimental data. (b) Detail of the primary amplification process. (Online version in colour.)

3.3. Model 2: deactivation but no recycling of amplifiers

In the second partial model, we use mechanisms (i), (ii) and (iv). We represent the lack of mechanism (iii) by setting

$$k_5=0.$$

All other parameters are the same as in table 1. This leads to a simplified model where

$$[J]\equiv 0,[G]\equiv 0.$$

The numerical simulation is shown in figure 6. The model also reproduces the amplitude of DNA production during the (II) primary amplification phase and the (III) plateau observed experimentally. This is evidence that the plateau is indeed caused by template deactivation. Note that the concentration of total DNA closely approximates the concentrations reported in EXPAR experiments. However, the model does not reproduce the (IV) secondary amplification phase. Since the only mechanism still missing is the (iii) recycling, we conclude that secondary amplification is caused by template recycling.

Figure 6.

Partial model 2 (k₅ = 0). (a) Total DNA and individual reaction products, shown with experimental data. (b) Detail of the primary amplification process. (Online version in colour.)

Without recycling, the deactivation of templates lowers the concentration of core amplification loops until trigger replication ceases completely. Total DNA growth after the core amplification loops are deactivated is due to dimerization of Y. When polymerase synthesizes over the toehold regions to create strongly associated inactive dimers D, the size of the strand is increased (figure 3), so total DNA continues to rise at a significantly diminished rate (note the decrease in Y and a corresponding increase in D in figure 6b).

3.4. Model 3: polymerase is not limiting

In the final partial model, we use mechanisms (i)–(iii), but not (iv), replacing the equation that models availability of polymerase, (2.1), with the assumption that polymerase stays at its initial level, (P₀), throughout the reaction. All other parameters of the model remain the same as in table 1. The numerical simulation is shown in figure 7.

Figure 7.

Partial model 3 ([P] ≡ P₀). (a) Total DNA and individual reaction products, shown with experimental data. (b) Detail of the primary amplification process. (Online version in colour.)

This model reproduces the (I) initiation, (II) primary amplification, (III) plateau and (IV) secondary amplification phases. However, the total DNA growth does not slow down at the rate observed in the experimental data. This suggests the reduced production of free trigger that drives the final deceleration is caused by competition for the available polymerase.

In this simulation, the secondary amplification phase is extended as inactive dimers form rapidly due to the abundance of the trigger. Again, because polymerase will synthesize over the toeholds of the triggers, one inactive dimer is larger than two triggers; see figure 3. As polymerase is not limiting, the core amplification loops continue to produce trigger which subsequently continues to form inactive dimers. This results in the linear growth that continues past the experimental time frame and leads to a significant increase in predicted total DNA production compared to experimental data. We expect that EXPAR templates would show the same linear growth without template inactivation. In light of this information, the fact that EXPAR experiments all show a plateau in total DNA production further supports the idea that the template deactivation is responsible for the termination of EXPAR amplification.

4. Discussion

In this paper, we propose the first mathematical model that reproduces biphasic behaviour of UDAR templates [11]. We posit that the necessary ingredients for successful reproduction of all five characteristic phases of these biphasic experiments must include gradual amplifier deactivation, recycling of deactivated amplifiers back into single-stranded templates and polymerase sequestration. Our model incorporates these three assumptions with parameters informed by experimental literature and successfully reproduces all essential features observed in UDAR trials. Furthermore, a sequence of partial models where some mechanisms are omitted failed to reproduce key aspects of the UDAR process.

The proposed mechanisms of template deactivation, template recycling and polymerase sequestration are not included in previous models of isothermal DNA amplification reactions [3,10,16]. Our results suggest that some of these mechanisms may also play a role in other isothermal DNA amplification processes, including the popular reaction EXPAR. Because EXPAR experiments produce final amplification levels comparable to the first plateau of a UDAR template, it is possible that the core amplification loops of EXPAR templates also stop replicating permanently due to template deactivation. Based on the presented model, template recycling plays the key role in the secondary amplification phase of UDAR templates. We expect that recycling of non-palindromic templates (such as those used in EXPAR) is energetically unfavourable and hence very rare. The palindromic binding sites make recycling more likely in two ways. First, triggers can separate deactivated complexes at two different locations on either strand, while for non-palindromic complementary sites trigger can only attach to the template and not to the complementary top strand. Second, the complementary top strand with palindromic sequences will form a hairpin loop. Because both the UDAR template and complementary top strand can close when separated, the strands are less likely to reassociate. Linear templates like those used in EXPAR, however, will have a strong affinity for their complementary top strand and are more likely to return to the deactivated state than be recycled.

It is worth noting that the biphasic behaviour observed in UDAR trials is not a requisite feature for the increased amplification levels. As noted in [11], a few templates fail to produce a plateau but do nevertheless attain much larger final concentrations than EXPAR templates. We expect that the trigger concentration required for recycling these templates is lower and therefore attained before template deactivation can noticeably slow the amplification. We further expect that templates that lack strong enough trigger associations to recycle amplifiers or that cannot produce the critical concentration of trigger will not fully enter the second amplification phase during the experimental time frame.

While this is the only comprehensive model to date incorporating polymerase sequestration as well as the only mathematical model describing a biphasic DNA amplification reaction, there are several limitations to be addressed in future work. First, a more detailed study of several different template types would help identify the relationship between DNA association thermodynamics with each reaction mechanism. Second, future studies will focus on identifying unknown specific parameters, such as an experimentally determined trimolecular rate constant (for dual trigger binding to a template) and dimerization, as both were only estimated by order of magnitude approximations. Finally, future work will include investigation of kinetic parameters k₅ and κ_R that describe the novel mechanism of template recycling and understanding the effect of reactions that were not included in the model. Future work will also identify mechanisms critical for type I templates, as well as identify potential simplifications that could occur for specific template types. Our model describes a basic reaction set that occurs during UDAR and informs mathematical studies of isothermal DNA amplification reactions such as EXPAR. The scope of this current work is to identify the critical reaction mechanisms required to reproduce biphasic amplification output. A mechanistic understanding of this reaction can potentially allow researchers to predict and control the kinetics and trigger output of UDAR based on known DNA association and dissociation kinetics. This kind of understanding will in turn produce more accurate models of isothermal amplification reactions. This model can be adapted to aid in the design of molecular diagnostics, DNA circuits and synthetic biology applications.

5. Methods

5.1. Reagents

SYBR Green II RNA Gel Stain was purchased from Thermo Fisher Scientific (Waltham, MA). Nuclease-free water, Oligo length standard 10/60 and desalted amplification templates (suspended in IDTE Buffer (pH 8.0) at a concentration of 100 μM and modified with an amino group on the 3^′ end to prevent template extension) were purchased from Integrated DNA Technologies, Inc. (Coralville, IA). Nt.BstNBI nicking enzyme, Bst 2.0 WarmStart DNA Polymerase, 10× ThermoPol I Buffer, dNTPs, bovine serum albumin (BSA) and 100 mM MgSO₄ were purchased from New England Biolabs (Beverly, MA). The template (LS3, 5^′-CGCGCGGTTTGGTAATGACTCTCGCGCGGTTTGG-3^′) and trigger (5^′-CCAAACCGCGCG-3^′) were previously described in [11]. The traditional EXPAR template design was taken from [12], with template (5^′-CTCACGCTACGGACGACTCTCTCACGCTAC-3^′) and trigger (5^′-GTAGCGTGAG-3^′).

5.2. UDAR set-up

Reactions were prepared at 4°C. The amplification reaction mixture contained 1× ThermoPol I Buffer (20 mM Tris–HCl (pH 8.8), 10 mM (NH₄)₂SO₄, 10 mM KCl, 2 mM MgSO₄, 0.1% Triton X-100), 25 mM Tris–HCl (pH 8), 6 mM MgSO₄, 50 mM KCl, 0.5 mM each dNTP, 0.1 mg mL⁻¹ BSA, 0.2 UμL⁻¹ Nt.BstNBI and 0.0267 UμL⁻¹ Bst 2.0 WarmStart DNA Polymerase (75 000 U mg⁻¹, 67 kDa). The amplification template was diluted in nuclease-free water and added at a final concentration of 100 nM. SYBR Green II (10 000 × stock in dimethylsulfoxide) was added to the reaction mixture to a final concentration of 4×. The experimental data shown in this work were triggered by 10 pM of trigger. Fluorescent measurements were taken every 20 s with a 12 s imaging step in a BioRad CFX Connect Thermocycler (Hercules, CA) at 55°C. Completed reactions were stored at −20°C for further analysis.

5.3. UDAR time series

To determine the conversion factor between the real-time fluorescence and the total DNA produced, 9 tubes of 20 μL UDAR reactions were produced according to the above protocol without the addition of trigger, which can be non-specifically produced during EXPAR and UDAR reactions [11]. One tube for each time point (0, 8.5, 10.2, 12.3, 19.4, 31.1, 60 min) was taken out of the thermocycler and placed on ice to terminate the reaction. The concentrations of the reaction products were determined using calibrated SYBR gold fluorescence of the 12 nucleotide trigger and used to calculate the μM/fluorescent unit. From these time points, we found that the conversion between fluorescence from the real-time thermal cycler and μM of products was 0.0026 ± 0.0004 μM/fluorescence unit after background correction. The number of nucleotides incorporated into the reaction products was calculated by multiplying the μM of measured reaction products by 12, as SYBR dye fluorescence is approximately correlated with length of the products and reaction products are largely composed of trigger Y. This conversion factor was used to calculate the concentration of reaction products in the trigger-initiated reaction described above.

Data accessibility

All data and the Matlab code used to numerically solve the ordinary differential equation model are included as electronic supplementary material.

Authors' contributions

D.C. created the Matlab model, drafted the manuscript and fitted the model and the data. T.G. and S.M. drafted the manuscript and conceived the model framework, and S.M. determined the appropriate ranges for kinetic parameters. B.O. produced the experimental data, created the figures and edited the manuscript.

Competing interests

S.M. and T.G. have a patent application on the biphasic DNA amplification method described here.

Funding

This work was supported by the Office of the Assistant Secretary of Defense for Health Affairs, through the Peer Reviewed Medical Research Program, Discovery Award under award no. W81XWH-17-1-0319. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the Department of Defense. T.G. was partially supported by NSF grant no. DMS-1361240, USDA 2015-51106-23970, DARPA grant no. FA8750-17-C-0054 and NIH grant no. 1R01GM126555-01. D.C. acknowledges the SMART Scholarship funded by the Department of Defense (US Army).