Jacques Arnaud

Laurent Chusseau

Fabrice Philippe

Fabrice Philippe On

research institutions in France or abroad, or from public or private research centers. On the ideal gas law

The air density on earth decays as a function of altitude z approximately according to an exp(-w z/θ)-law, where w denotes the weight of a nitrogen molecule and θ = kBT where kB is a constant and T the thermodynamic temperature. To derive this law one usually invokes the Boltzmann factor, itself derived from statistical considerations. We show that this (barometric) law may be derived solely from the democritian concept of corpuscles moving in vacuum. We employ a principle of simplicity, namely that this law is independent of the law of corpuscle motion. This view-point puts aside restrictive assumptions that are source of confusion. Similar observations apply to the ideal-gas law. In the absence of gravity, when a cylinder terminated by a piston, containing a single corpuscle and with height h has temperature θ, the average force that the corpuscle exerts on the piston is: F = θ/h. This law is valid at any temperature, except at very low temperatures when quantum effects are significant and at very high temperatures because the corpuscle may then split into smaller parts. It is usually derived under the assumption that the temperature is proportional to the corpuscle kinetic energy, or else, from a form of the quantum theory. In contradistinction, we show that it follows solely from the postulate this it is independent of the law of corpuscle motion. On the physical side we employ only the concept of potential energy. A consistent picture is offered leading to the barometric law when w h ≫ θ, and to the usual ideal-gas law when w h ≪ θ. The mathematics is elementary. The present paper should accordingly facilitate the understanding of the physical meaning of the barometric and ideal-gas laws.

Introduction

The purpose of this paper is to show that the barometric law and the ideal-gas law may be obtained on the sole basis of the democritian model according to which nature consists of corpuscles moving in a vacuum, plus a principle of simplicity: namely that these fundamental laws are independent of the law of corpuscle motion (non-relativistic, relativistic, or otherwise: see appendix D). The temperature θ enters into the ideal-gas law and the expression of the gas internal energy solely for dimensional reasons. We show from the general expressions of the gas internal energy and of the force (or pressure) that the heat delivered by the gas is θ dS, an expression of the entropy S being given. This result enables us to prove that the formally-introduced temperature θ coincides, to within some arbitrary constant factor, with the thermodynamic temperature T . Indeed, we recover for ideal gases the general Carnot result asserting that the maximum efficiency of thermal engines is: 1 -θ l /θ h , where θ l denotes the cold bath temperature and θ h the hot bath temperature. The reader may feel that our statement that the above invariance principle implies the barometric and ideal-gas laws, without anything else, is quite surprising. Yet, we hope that we can convince him/her that this is indeed the case.

Let us emphasize that our goal is to derive the barometric and ideal-gas laws from first principles, only conservation of potential energy being assumed. We do not use the concept of kinetic energy, nor do we postulate any law of corpuscle motion. Accordingly, a given potential φ(z) = w z for a weight w does not imply any specific law of motion z(t).

Concepts relating to heat since the antiquity: Democritus, who lived about 300 years B.C., described nature as a collection of corpuscles that cannot be split, moving in vacuum. These corpuscles differ from one-another in form, position and weight. In the case of a gas, interaction between corpuscles may often be neglected, but they collide with the container's walls. Platon [1] ascribed heat to corpuscular motion: "Heat and fire are generated by impact and friction, but that's motion". Much later, Francis Bacon (1561-1626) wrote: "The very nature of heat is motion, and nothing else". This view-point is more explicit in Daniel Bernoulli writing (1738): "Gas atoms are moving randomly, and pressure is nothing else but the impact of the atoms on their container walls". Lastly, Carnot introduced energy considerations circa 1830: "Heat is nothing but motive power, or rather another form of motion. When motive power is destroyed, heat is generated precisely in proportion of the motive power destroyed. Likewise, when heat is destroyed, motive power is generated ". We will employ the law of conservation of potential energy, well known since the antiquity from cords and pulleys experiments.

Experimental results relating to air: The first accurate experiments relating to gases are tied up to the invention of the thermometer by Galileo and the barometer by his assistant Torricelli. Then, to Pascal experiments on atmospheric pressure. Pascal ascribed the diminution of the height of a mercury column as a function of altitude to the reduction of the weight of the air above the barometer. It was later shown that the pressure decays exponentially.

Let us recall the crucial experiments performed in the seventeenth century concerning the properties of air. Air, consisting mostly of di-atomic nitrogen, may be viewed as an ideal gas. When a tight box contains some amount of air, the volume-pressure product is a constant at room temperature, a law enunciated for the first time by Boyle in 1660: "Pressure and expansion are in reciprocal proportions". Boyle employed a J-shaped glass tube, with the sealed small side full of air, and the other full of mercury. The left-side height was a measure of volume and the right-side height a measure of pressure. Subsequent experiments have shown that this law is applicable at any constant temperature, for example at various liquid boiling temperatures, within some experimental range. We call "generalized Boyle law" the expression: V(P, θ) = f (θ)/P, where P represents pressure, V volume, and f (θ) some temperature measure.

From an experimental standpoint we could define temperature as the pressure relating to some given amount of matter contained within some fixed volume. As the temperature gets higher the pressure increases. This pressure may be used to define θ. Of course, different temperature scales would be obtained for different substances, but such thermometers may be calibrated one against another because temperatures tend to equalize in equilibrium. Rarefied helium may be described with great accuracy as a collection of independent corpuscles, except perhaps at very low temperatures when quantum effects become significant and at very high temperatures when the helium atoms may get ionized. The theory presented in this paper shows on the basis of the corpuscular model that θ, initially introduced formally from dimensional considerations, coincides with the thermodynamic temperature. This is the temperature that enters in the expression of thermal-engine efficiencies.

Gay-Lussac has shown in 1802 that, at atmospheric pressure, the volume increment of various gases from freezing to boiling water temperatures is 37.5% [START_REF] Holbrow | What Gay-Lussac didn't tell us[END_REF]. Appropriate gas thermometers enabled experimentalists to establish the proportionality of volume and temperature at constant pressure. This measurement was subsequently made at various pressures, for exemple at various altitudes. The generalized Gay-Lussac law may be enunciated as follows: The two-variable function V(P, θ) = θ g(P), where g(P) is some unknown function of pressure. Comparison of the generalized Boyle and Gay-Lussac laws shows that: P V = θ h(N ), where h(N ) defines the amount of gas considered.

Let us emphasize that the empirical Gay-Lussac law makes sense only if one specifies which thermometer is employed. One may employ a gas thermometer from a selected gas such as helium in two ways. One method consists of defining the temperature as the cylinder height (or volume) at a fixed pressure, for example at the standard atmospheric pressure. In the following, we assume that a second method is being employed instead: the temperature is defined as the force that must be exerted on the piston to maintain the height at a fixed value, for example one meter, as said above. If the Gay-Lussac experiment were applied to a gas identical to the gas employed in the thermometer (helium in our example) the fact that pressure is proportional to temperature would be obvious. The importance of the Gay-Lussac experiment is that the proportionality law is found to be valid for any gas.

However, it was subsequently discovered that the Gay-Lussac proportionality law is reasonably accurate only at very small pressures. The theoretical reason that explains this observation is that, at low pressures, the gas molecules of the tested gas and those of the thermometer gas may both be considered as independent non-interacting corpuscles (see below).

In 1803, Dalton, on the basis of his studies of chemical compounds and gaseous mixtures suggested that matter consists of atoms of different masses that combine in simple ratios. He discovered the partial-pressure law according to which the total pressure exerted by a gas mixture is equal to the sum of the pressure that each one of the gases would exert if it occupied the full volume alone. Finally, in 1811, Avogadro concluded that equal volumes of gases at the same temperature and pressure contain the same number of molecules (or corpuscles). This entails that P V/θ is proportional to N , now interpreted as the number of corpuscles. One calls "Avogadro number", N A , the number of corpuscles contained in 0.0224 cubic meters of gas in standard conditions. In 1865, Loschmidt established from a measurement of the air viscosity that N A is on the order of 10 23 . Many other methods have been employed since then for that purpose, such as Brownian motion.

On empirical grounds, the ideal-gas law may therefore be written as:

P V = N θ (1)
where V denotes the volume, P the pressure, N the number of corpuscules, and θ ≡ k B T the temperature.

The ideal-gas law has been partly explained on the basis of a kinetic theory by Waterston [START_REF] Waterston | On the physics of media that are composed of free and perfectly elastic molecules in a state of motion[END_REF] in 1843, the kinetic theory being based itself on non-relativistic mechanics. The next important theoretical discovery is due to Boltzmann, see Section 2. In subsequent sections we recall the basic assumptions on which rest the usual proofs of the barometric and ideal-gas laws. Then we present our model.

Usual kinetic and statistical theories

A recent reference [START_REF] Miller | Ideal gas law. Mauricius[END_REF] lists the assumptions on which the gas kinetic theory rests. Some of them express the democritian hypothesis and are indeed essential. The usefulness of the others, listed below, however, may be questioned:

1. Gases consist in corpuscles having non-zero mass.

2. The corpuscles are quickly moving.

3. They are perfectly spherical and elastic.

4. The average kinetic energy depends only on the system temperature.

5. Relativistic effects are negligible. 6. Motion laws are time reversible.

7. The number of molecules is so large that a statistical treatment is appropriate.

Comment: As we shall show, none of the above assumptions are needed. It suffices that the (perhaps unique) corpuscle be in thermal contact with the ground.

We sketch in the present section the most usual derivations of the barometric and ideal-gas laws to remind the readers of the underlying assumptions. Note that the barometric law may be obtained from the ideal-gas law, and conversely, if one postulates that weightless plates may be introduced or removed at will in the gas at various altitudes. But this postulate is at best plausible. In an interesting paper, Norton [START_REF] Norton | Atoms, entropy, quanta: Einstein's miraculous argument of 1905[END_REF] derived the ideal-gas law from the barometric equation. However, the latter involves the Boltzmann factor, which requires other physical considerations (see below), while in the present paper this factor comes in naturally, that is, for purely mathematical reasons.

The barometric law is usually viewed as a straightforward consequence of the Boltzmann factor: the probability that a corpuscle has energy E is proportional to: exp(-E/θ) where θ = k B T and the energy E = w z, where w denotes the corpuscle weight (e.g., the weight of a di-atomic nitrogen molecule) and z the altitude. Hence the exponential decay.

The derivation of the Boltzmann factor itself is based on a quantization of the energy 1 , and the postulated equi-probability of the micro-states. Let the discrete (non-degenerate) energy levels be denoted by ε 1 , ε 2 , .... If distinguishable corpuscles are distributed among the energy levels, with n 1 corpuscles in level 1, n 2 corpuscles in level 2, and so on, the number of ways of doing that is inversely proportional to: n 1 ! n 2 !.... It is postulated that this number reaches its maximum value at equilibrium under the constraint that n 1 + n 2 + ... = N , the total number of corpuscles, and n 1 ε 1 + n 2 ε 2 + ... = E, the total energy. In the limit of large N values, one finds that: n i ∝ exp(-ε i /θ), for some θ-value that depends on N and E. Even though physicists are now-a-day very familiar with that procedure, it is not so easy to explain it to students. Besides, it rests on a number of assumptions.

The traditional derivation of the ideal-gas law, on the other hand, is based on non-relativistic mechanics [START_REF] Feynman | Feynman lectures in physics; mainly mechanics, radiation and heat[END_REF]. For a one-dimensional model, one considers a corpuscle moving back and forth between two plates separated by a distance h, one of them playing the role of a piston. If v denotes the speed of a corpuscle, an impact on a plate delivers to it an impulse 2 m v where m denotes the corpuscle mass, and this occurs every 2h/v time units. It is concluded that the force F exerted on the piston is:

2 m v/(2h/v) = m v 2 /h, that is: F h = 2 K, where K = 1
2 m v2 denotes the kinetic energy. It is recognised that there may be a distribution of kinetic energies [START_REF] Feynman | Feynman lectures in physics; mainly mechanics, radiation and heat[END_REF]. Postulating that the temperature θ is proportional to the average kinetic energy one obtains for the average force the ideal-gas law: F h ∝ θ. Alternatively, one may quantize the corpuscle wave-function and employ the Bolzmann factor [START_REF] Arnaud | Démocrite et la puissance motrice du feu (Democritus and the motive power of fire)[END_REF].

The procedure described above has been generalized to relativistic motion (k B T ∼ mc 2 ). The same ideal-gas law is valid at any temperature (within the corpuscular model). Our thesis is that the ideal-gas law has simply nothing to do with the law of corpuscle motion, and that it is therefore not surprising that it holds for both the Galileo and Einstein laws of motion. The assumption that temperature is proportional to the average kinetic energy cannot possibly be derived from first principles since it is only an approximation acceptable when k B T ≪ mc 2 . These are some of the reasons why we feel that the traditional proofs are unsatisfactory. An alternative is offered below.

The barometric law

We are considering an unit-area cylinder with vertical axis in uniform gravity, resting on the ground (z = 0) at some temperature. We consider only motion along the vertical z-axis, denoted in general by z = z(t; E), where t denotes time. The corpuscle energy is defined as: E ≡ w z m where w is the corpuscle weight and z m the maximum altitude. In the following, some regularity of the z(t)-function is assumed, but no specific law is presumed, except in examples. We set for convenience t = 0 at the top of the trajectory, that is: z(0) = z m , z ′ (0) = 0, where a prime denotes a derivative with respect to t.

Consider a single period of corpuscle motion as shown in Fig. 1. Let the corpuscle distance from the top of its trajectory be denoted Z ≡ z m -z ≥ 0, at times t 1 and t 2 ≥ t 1 . We call 2 : "time interval" τ (Z) ≡ t 2 -t 1 . Because gravity is static and uniform (that is, independent of altitude and time) this τ -function depends only on Z. As an example, for non-relativistic motion: Z(t) = 1 2 g t 2 , with w ≡ m g where m denotes the corpuscle mass and g the gravitational acceleration. In that example: τ (Z) = 2 2 Z/g ∝ √ Z, g being a constant. The period of motion of a corpuscle bouncing off the ground (z = 0) without any loss of energy (rigid walls and negligible gas friction), and having energy E ≡ w z m , is according to the above definitions: τ (z m ). On the other hand, the time spent by the corpuscle above some z-level is obviously zero if z > z m , and equal to:

τ (Z) ≡ τ (z m -z) if z ≤ z m .
In the latter case, the Figure 1: The curve represents the motion z = z(t) of a corpuscle of weight unity submitted to static uniform gravity. Note that the results presented in this paper do not depend on the law of motion, which needs not be an even function of time. (The particular curve shown refers to a relativistic law of motion based on a clock synchronisation different from the usual (Einstein) one. Namely, we suppose that clocks at various altitudes z are synchronized by a light pulse emitted from z = 0 at t = 0. Then the apparent speed of light is infinite upward and equal to c/2 downward. The asymptote is ζ ≡ z -z m = -1/2t. We give these details because the reader may be puzzled by the curve represented on this figure. However, almost any other curve would serve our purposes as well). The maximum height z m reached above the ground level z = 0 is the corpuscle energy E. τ (z m ) represents the motion period. A reflecting plane (piston) is shown at the height z = h as a dotted line, the corpuscle bouncing alternately off the two planes. From a simple inspection of the figure on sees that the period becomes in that case:

τ (z m ) -τ (z m -h).
fraction of time during which the corpuscle is above z is therefore: τ (z m -z)/τ (z m ), as suggested on the figure 1.

We now suppose that the ground on which the corpuscle is bouncing off has been heated to some temperature (the concept of temperature will be precisely defined later on). This means that the ground level (z=0) is not perfectly static as assumed above, but instead is quivering. Concretely, the groung level could be moving up and down according to some zero-mean fluctuation of small amplitude so that, upon impacting on the ground, the corpuscle may loose energy (when the ground level is moving downward), or gain energy (when the ground level is moving upward). We will not need the explicit form of this fluctuation. We only assume that the amplitude of that fluctuation is so small that the corpuscle energy does not vary significantly over many periods. Yet it may evolve slowly. The energy distribution ω(E) refers to averages over arbitrarily long times, and is presently unknown; it will be determined by enforcing the condition that the law of interest (presently the barometric law) does not depend on the corpuscle law of motion.

The fractional time during which the corpuscle is above some z-level is, according to the previous considerations and remembering that z m = E/w where w is a constant:

A(z) = ∞ wz dE ω(E) τ (E/w -z)/τ (E/w) ∞ 0 dE ω(E)
.

(

) 2
The lower limit of the integral in the numerator is wz since the fractional time is equal to zero when E ≤ wz.

We now argue that ω(E) must be: exp(-E/θ)τ (E/w), where θ is a constant having the dimension of energy as is required by the fact that the argument of an exponential is dimensionless. First, let us introduce this distribution law in [START_REF] Holbrow | What Gay-Lussac didn't tell us[END_REF]. We obtain:

A(z) = ∞ wz dE exp(-E/θ)τ (E/w -z) ∞ 0 dE exp(-E/θ)τ (E/w) = exp(-w z/θ) ∞ wz dE exp(-(E -wz)/θ)τ (E/w -z) ∞ 0 dE exp(-E/θ)τ (E/w) = exp(-w z/θ) ∞ 0 dE ′ exp(-E ′ /θ)τ (E ′ /w) ∞ 0 dE exp(-E/θ)τ (E/w) = exp(-w z/θ). (3)
On the third line, we have used as an integration variable E ′ ≡ E -wz instead of E. The last line follows from the fact that E, E ′ are dummy variables, so that we may replace E ′ by E. Note that, even though we have introduced integral signs, no integral has been performed. The distribution ω(E) introduced above is the one that leads to a result (last line of (3)) independent of the particular form of the τ (Z)-function, and therefore of the law of motion. For a purely mathematical reason, the term: τ (E/w) must be there to cancel a similar term in the denominator of the expression of the fractional time. On the other hand, the only function of u ≡ E/θ that may cause the integral in the numerator to go from 0 to infinity and cancel out with the integral in the denominator is: exp(-u), the argument being defined only to within a constant factor. In order to obtain the energy distribution ω(E), one would need to know the function τ (Z). But, remarkably, the energy distribution is not needed explicitly.

From a physical standpoint, the energy distribution may be written as exp(-E(f )/θ), where f denotes the action, equal to the z(t) motion area for one period, and df /dE = τ (E/w). In quantum mechanics the action f is set equal to an integer (1,2..., ignoring a small constant) times the reduced Planck constant . The term exp(-E/θ), usually referred to as the "Boltzmann factor", enters here solely by mathematical necessity [START_REF] Arnaud | Démocrite et la puissance motrice du feu (Democritus and the motive power of fire)[END_REF].

The gas density, defined as the probability that the corpuscle be located between z and z +dz, divided by dz, is: ρ(z) = -dA(z)/dz = (w/θ) exp(-wz/θ). This is the barometric law. Since w and θ are constant, the density decays exponentially as a function of altitude z. In the earth atmosphere the density of di-atomic oxygen decays faster than the density of di-atomic nitrogen because the weight of an oxygen atom exceeds that of a nitrogen atom in the ratio ≈ 16/14.

We have supposed above that the weight w is independent of altitude. A more general formulation is given in appendix.

Potential energy

We will only need the gas internal energy U . However, it is of some pedagogical interest to separate out from U a term that may be called the potential energy. This term is directly calculated in the present section. The potential energy of a corpuscle of weight w located at altitude z is defined as w z. The gas potential energy, according to the above expression of the density, is therefore:

P = w ∞ 0 dz z ρ(z) = w ∞ 0 dz A(z) = θ, using (3).
When the corpuscle motion is restricted to: 0 < z ≤ h one can show that the density ρ(z) is unaffected in that range of z, as is discussed in an appendix. It must be normalized, though, so that the integral of ρ(z) from 0 to h be unity. A simple integration of w z ρ(z) from z = 0 to z = h then gives the potential energy for any h-value:

P = θ - w h exp(w h/θ) -1 . (4)
According to that expression P tends to θ monotonically if w h → ∞. It tends to 0 when w h tends to 0, that is in the absence of gravity. This is what is usually assumed when one refers to the ideal-gas law. In that case, as shown below, the internal energy depends only on temperature.

Internal energy

The gas internal energy U is the average value of E, if only motion along the z-axis is considered (note that the gravitational energy is accounted for in U ). It can be evaluated by the above method and is found to be given by the sum of the potential energy in (4) and a term that depends on θ but not on h. The explicit expression of U ≡ E is, omitting details that are similar to those in section 3:

U = w h 0 dE E exp(-E/θ)τ (E/w) + ∞ w h dE E exp(-E/θ) τ (E/w) -τ (E/w -h) w h 0 dE exp(-E/θ)τ (E/w) + ∞ w h dE exp(-E/θ) τ (E/w) -τ (E/w -h) = P + K, (5)
where the potential energy P is given in (4) and:

K = ∞ 0 dE E exp(-E/θ)τ (E/w) ∞ 0 dE exp(-E/θ)τ (E/w) -θ (6)
depends only on temperature. The explicit form of K requires the knowledge of the τ (Z)-function and therefore of the corpuscle law of motion. In the special case of non-relativistic motion, for example, we have τ (Z) ∝ √ Z, see Section 3. At that point, Integration is needed. It gives: K = θ/2, or more precisely: K = k B T /2, a well-known result. In Physics text-books K is called the average kinetic energy. This interpretation, however, is not needed here: K is simply the term that remains when the potential energy is subtracted from the internal energy.

Force exerted on the piston by a corpuscle of energy E

We now treat the ideal-gas law by similar methods. We are considering again a unit-area cylinder with vertical axis in uniform gravity, resting on the ground (z = 0) at some temperature. A tight piston can move in the vertical z direction. The cylinder height is denoted by h and contains a single corpuscle of weight w. In our one-dimensional model, the pressure P corresponds to the average force F , the volume V to the height h, and N = 1. Our result provides the ideal-gas law in a generalized form, taking into account gravity. In that case, the pressure varies as a function of altitude. More precisely, the force exerted by the corpuscle on the lower end of the cylinder exceeds the force exerted on the upper end (or piston) by the corpuscle weight. But in the absence of gravity, the forces exerted on both ends are the same.

We are introducing (static and uniform) gravity, not so much for the sake of generality, but because this helps clarify the concept of corpuscle energy: the corpuscle energy is defined as the maximum altitude that the corpuscle would reach in the absence of the piston, multiplied by the corpuscle weight. Precisely, the maximum height reached by a corpuscle of weight w and energy E in the absence of a piston is: z m ≡ E/w, the corpuscle bouncing elastically from the ground, that is, without any loss or gain of energy.

Consider first the case where h is infinite, that is, in the absence of a piston. The time period is denoted as before τ (Z) with Z = z m . The average force exerted on the ground, equal to the corpuscle weight w, is the product of the impulse i and the number of impulses per unit time. Thus w = i/τ (z m ) -→ i = w τ (z m ). In other words, the impulse transmitted to a plane when the corpuscle impacts on it is the product of the corpuscle weight and the motion period.

If the plate is located at z = h, we have Z = z -z m the impulse is: i h = w τ (z m -h). When the corpuscle is moving back and forth beween the planes at z = 0 and z = h (instead of being located above h) the impulse is just opposite to i h . It is therefore in absolute value equal to w τ (z m -h).

Next, we introduce a rigid plane at z = h, viewed as a piston, and consider a corpuscle bouncing on the z = 0 and z = h planes alternately. The time period becomes: τ (z m )-τ (z m -h), as one can see from inspection of the figure. We call F the force exerted on the z = h rigid plane, averaged over a time period. It follows from the previous expressions that:

F = 0 z m ≤ h F = i h τ (z m ) -τ (z m -h) = w τ (z m -h) τ (z m ) -τ (z m -h) z m > h. ( 7
)

Average force

As a consequence of the slight quivering of the cylinder lower end (thermal motion), the corpuscle energy E slowly varies in the course of time. The force F just defined must be weighed by some energy distribution ω(E) in such a way that the average force F be independent of the corpuscle equation of motion, and thus of the τ (.)-function. In the limit where w h ≪ θ, a condition that amounts to ignoring gravity, we obtain the ideal-gas law in the form: F = θ/h, where θ is a quantity having the dimension of an energy. We later on prove that θ is the thermodynamic temperature.

The above condition obtains from [START_REF] Arnaud | Démocrite et la puissance motrice du feu (Democritus and the motive power of fire)[END_REF] if and only if one selects the following energy distribution:

ω(E) = exp(-E/θ)τ (E/w) E ≤ w h ω(E) = exp(-E/θ) τ (E/w) -τ (E/w -h) E > w h, (8)
where θ has the dimension of an energy. Indeed, the average force becomes, using ( 7) and ( 8):

F = w ∞ w h dE exp(-E/θ)τ (E/w -h) w h 0 dE exp(-E/θ)τ (E/w) + ∞ w h dE exp(-E/θ) τ (E/w) -τ (E/w -h) = w exp(w h/θ) -1 . ( 9
)
In the above integrals going from w h to ∞ we have replaced exp(-E/θ) by exp(-w h/θ) exp(-(Ew h)/θ) and introduced the variable E ′ ≡ E -w h, so that all the integrals go from zero to infinity and cancel out. Note that, as in Section 3, no integration has been actually performed. The corpuscles being independent, for an arbitrary collection of N corpuscles having the same weight the force is multiplied by N . In the case of zero gravity w=0 or more precisely: w h ≪ θ. The above expression then gives: F h = θ. Thus we have obtained the ideal-gas law:

F h = N θ.
The expressions given earlier for the average force F in (9) and the internal energy U in (5) may be written, setting β ≡ 1/θ, as:

F = ∂ ln(Z) β ∂h ≡ - ∂A ∂h U = - ∂ ln(Z) ∂β ≡ A -θ ∂A ∂θ Z(β, h) = exp(-β w h) -1 ∞ 0 dE exp(-β E)τ (E/w). (10)
Z is essentially the quantity called in thermodynamics the partition function. It becomes dimensionless if it is divided by the reduced Planck constant , which however plays here no physical role. The continuous energy E in the integral may be replaced by closely-spaced discrete energies ε i , i = 1, 2..., the spacing between adjacent energies accounting for the τ -function. This procedure is the one employed in the numerical evaluation of integrals; it does not in itself implies quantization. The factor preceding the integral in (10) entails that the energies ε i , i = 1, 2... are multiplied by some function of h. In the second expressions of F and U given in (10) we have introduced for brevity the so-called "free energy" A(θ, h) ≡ -θ ln(Z(θ, h)).

To be sure, the present paper does not provide explicit expressions of gases internal energy if the law of corpuscle motion remains unknown. It does provide, however, a first-principle proof of the ideal-gas law, including a possible effect of uniform gravity, and the barometric equation. We have recovered the usual thermodynamical and semi-classical statistical-mechanical expressions for the special case of ideal gases.

8 The energy θ is the thermodynamic temperature

We now prove that θ, introduced in the previous sections on dimensional grounds only, is the thermodynamic temperature. We do this by showing that the maximum efficiency of a thermal cycle employing ideal gases is: 1 -θ l /θ h , where θ l is the cold-bath temperature and θ h the hot bath temperature.

From (10) we obtain:

F = - ∂A ∂h U = A -θ ∂A ∂θ -δQ ≡ dU + F dh = dA - ∂A ∂θ dθ - ∂A ∂h dh + θ dS = θ dS S ≡ - ∂A ∂θ , (11)
where δQ represents the heat released by the gas, from the law of conservation of energy. For any function f (θ, h) such as U, A, S: df ≡ ∂f ∂θ dθ + ∂f ∂h dh. If the gas is in contact with a thermal bath (θ= constant), δQ is the heat gained by the bath. The quantity S defined above is called the "entropy". In particular, if heat cannot go through the gas container wall (adiabatic transformation) we have δQ = 0 that is, according to the above result: dS = 0. Thus adiabatic transformations are isentropic. Note that introduction of A and S is only a matter of mathematical convenience.

A Carnot cycle consists of two isothermal transformations at temperatures θ l and θ h , and two intermediate adiabatic transformations (dS = 0). After a complete cycle, the entropy recovers its original value and therefore dS l + dS h = 0. According to (11): -δQ l = θ l dS l , -δQ h = θ h dS h and therefore δQ l /θ l + δQ h /θ h = 0. Energy conservation gives the work performed over a cycle from: W + δQ l + δQ h = 0. The cycle efficiency is defined as the ratio of W and the heating -δQ h supplied by the hot bath. We have therefore: η ≡ W -δQ h = δQ h +δQ l δQ h = 1 -θ l θ h , from which we conclude that θ is the "thermodynamic temperature". This temperature is defined only to within a multiplicative factor, which is fixed by agreeing that the water triple-point temperature is 273.16 kelvins. One thus sets: θ ≡ k B T . One generally considers an amount of gas called a "mole" occupying a volume of 0.0224 cubic meters at standard pressure and temperature (approximately one atmosphere or 100 000 pascals, and 300 kelvins). We then write: P V = R T , with the ideal-gas constant: R ≈ 8.314... joules per kelvin.

Conclusion

Let us briefly recall the concepts introduced in the present paper. One can imagine that after having introduced the corpuscular concept, Democritus observed the elastic bounces of a unit weight on a balance and defined the weight "impulse" from the motion period. Not knowing the nature of the motion (parabolic? hyperbolic?), he may have thought of introducing a weight factor such that the average force F does not depend on the law of motion. This, as we have seen, may be done. This weight factor involves for dimensional reasons a quantity θ having the dimension of energy. Considering a thermal engine operating between two baths at temperatures θ l , θ h one finds on the basis of the principles just stated that the maximum efficiency is: 1-θ l /θ h . This allows us to call θ the thermodynamic temperature.

William of Ockham (1287-1347) set as a matter of principle that one should not employ more concepts than those that are strictly necessary to explain the observed phenomenas. (Some authors consider that the Ockham philosophy predates the advent of modern science by insisting on facts and the kind of reasoning employed rather than on speculations about essences). Accordingly, it seems important to elucidate the assumptions on which rest, in particular, the barometric and ideal-gas laws that play an essential role in theoretical physics and many applications. Our thesis is that these laws may be obtain on the sole basis of the Democritus model of corpuscles and vacuum. It is indeed unnecessary to specify the laws of motion. One can also show that the ideal-gas internal energy depends only on temperature (in the absence of gravity). To evaluate explicitly this energy it is, however, necessary to know the law of motion. From a pedagogical standpoint and in application of Ockham's concept one should not postulate principles which, without being erroneous, are unnecessary.

A A simple but incomplete proof of the ideal gas law

We present in this paragraph a simple proof leading to the ideal-gas law. Initially, we only suppose that the gas corpuscles are independent, so that the force exerted on the piston by N corpuscles is N times the force exerted by a single corpuscle, the other conditions (temperature and cylinder height) being the same. We also postulate that the force exerted by a corpuscle of any kind (e.g., nitrogen or helium) on the piston depends only on temperature and height (or volume). At the end of the argument, we additionally postulate that intermediate pistons may be removed without affecting the system equilibrium. This latter assumption would be untenable if the corpuscles were attracting each others, as is the case for non-ideal gases. The present proof is incomplete because we postulate, rather than demonstrate, that plates may be added or removed at various altitudes without resulting into any physical effects.

Let us consider a cylinder of unit area, with a tight piston that can move freely along the axis. This cylinder of height h contains N corpuscles and is raised at some temperature θ. The cylinder may, for exemple, be filled with nitrogen at standard temperature and pressure. Because the corpuscles are independent, N corpuscles exert N times as much pressure as a single corpuscle, θ and h being unchanged. We can therefore set in general: F = N f (θ, h), where f (., .) is an unknown two-variables function. We can (at least in principle) define θ as the force that a single helium atom exerts on the piston when the cylinder height h = 1. Then: F = f (θ, 1).

If we now superpose h such cylinders, possibly containing corpuscles of various kinds, the total height becomes h and the number of corpuscles becomes h also. The force F exerted on the upper piston gets transmitted unchanged to each of the cylinders, if we neglect the gas and cylinder weight. This amounts to saying that each cylinder remains in the same conditions as before. But the system presently considered has height h, contains h corpuscles, and the force is: F = θ. Substituting these values of h, N and F in the general expression: F = N f (θ, h), we get: θ = h f (θ, h). Thus f (θ, h) = θ/h, and the general formula becomes: F = N θ/h, which is the ideal gas law.

We have implicitly assumes above that one can remove the intermediate pistons without modifying the system equilibrium; this is plausible if the corpuscles do not interact. A proof is given in Appendix B.

B Barometric equation with a plate

We consider the case where the corpuscle of weight w is bouncing between the ground z = 0 and a fixed plate at z = h. We are seeking the density: ρ(z o ), where 0 < z o < h. In Section 4, we have supposed that this density is the same as if the plate were not there (except for a normalization factor). Even though this assumption is plausible, it is useful to verify it to prove our formalism consistency. As in the main text we denote by z m the maximum height that the corpuscle would attain if the plate were not there, as a consequence of its energy. We set w = 1, θ = 1 for brevity.

To evaluate the probability p that the corpuscle be below z o , one must distinguish three cases:

0 < z m < z o : p = 1 z o < z m < h : p = τ (z m ) -τ (z m -z o ) τ (z m ) z m > h : p = τ (z m ) -τ (z m -z o ) τ (z m ) -τ (z m -h) (12)
On the other hand, the weighting factors relating to z m given in (8) are:

z m < h : exp(-z m )τ (z m ) z m > h : exp(-z m )(τ (z m ) -τ (z m -h)) (13)
The probability that the corpuscle be below z o is therefore:

zo 0 dz m exp(-z m )τ (z m ) + h zo dz m exp(-z m )(τ (z m ) -τ (z m -z o )) (14) + ∞ h dz m exp(-z m )(τ (z m ) -τ (z m -z o )) = (1 -exp(-z o )) ∞ 0 dz m exp(-z m )τ (z m ). ( 15
)
After normalization, we see that the probability that the corpuscle be below z o is: 1exp(-z o ), which is the expected result if the density is the same as in the absence of the plate at z = h, namely: exp(-z o ).

C Non-uniform gravity

In the present appendix we show how non-uniform weights w(z) ≡ m g(z), or continuous potentials: φ(z) ≡ z 0 dz w(z), could be handled. On earth, weights decay in proportion to the reciprocal of the square of the distance from the earth center. It is only for small changes in altitude that weights may be considered constant. Another example is a cylinder rotating about its axis at some constant angular rate. In that case g is proportional to the distance from axis.

We only treat the case of two weight values: w 1 , 0 < z ≤ h and w 2 , z > h, and evaluate the ratio of the average times spent by the corpuscle below and above the h altitude. Besides energy conservation we only suppose that corpuscle speeds are continuous. As in the main text, the corpuscles considered are bouncing off the ground at z = 0, at some temperature θ.

Let us denote by τ i (Z) the time-interval corresponding to a distance Z from the top of the trajectory when the weight is a constant w i , with τ i (0) = 0. These functions are of course different for different weight values. Considering two weight-values corresponding to subscripts 1,2, continuity of the corpuscle speed entails that: τ 2 (Z) = (w 1 /w 2 )τ 1 (w 2 Z/w 1 ), as one can see by taking the derivative of this relation with respect to Z. This relation is readily verified for the special case of Galilean motion in which case: τ 1 (Z) = C Z/w 1 , τ 2 (Z) = C Z/w 2 for a constant C. As in the main text we will let E denote the corpuscle energy.

When E ≤ w 1 h the corpuscle stays below the h-plane. When E > w 1 h, the time spent by the corpuscle above the h-plane is: τ 2 ( E-w1 h w2 ) because the corpuscle weight is w 2 and its energy with respect of the h-plane is E -w 1 h. The time spent below the h-plane is in that case: τ 1 ( E w1 ) -τ 1 ( E w1 -h). Thus the average times spent below and above the h-plane are respectively:

average time below =

If ω(E) denotes the energy distribution, the ratio of the average times spent by the corpuscle below and above h is therefore:

T = ∞ 0 dE ω(E)τ 1 (E/w 1 ) - ∞ w1 h dE ω(E)τ 1 (E/w 1 -h) (w 1 /w 2 ) ∞ w1 h dE ω(E)τ 1 (E/w 1 -h) (17)
The only way to remove the τ -functions is to choose: ω(E) = exp(-E/θ), where θ has the dimension of an energy. Proceeding as in the main text, we obtain from the above expression the result:

T = w 2 w 1 exp( w 1 h θ ) -1 . (18)
We have therefore obtained a result applicable to non-linear potentials. A possible generalization would consist of considering arbitrary static potentials. This, however, will not be done here. The expression in (18) is usually obtained by postulating a gas density: ρ(z) ∝ exp -φ(z)/θ , where φ(z) is the potential at z. T is now viewed as the ratio of the integral of ρ(z) from 0 to h and its integral from h to ∞. We have: φ(z) = w 1 z, z ≤ h and φ(z) = w 1 h + w 2 (z -h) = w 2 z + (w 1 -w 2 ) h, z > h.

This concept was introduced by Boltzmann who, however, considered only the limit in which the difference between successive energies is arbitrarily small.

We do not assume that z(t) = z(-t) or t 2 + t 1 = 0. Asymmetric laws of motion occur if one employs clock synchronisation rules different from the one proposed by Einstein. For example, if a light pulse emitted from z = 0 at t = 0 propagating upward is employed to synchronise clocks located at different altitudes, the apparent upward speed of light is, by this convention, infinite. The downward speed of light is then c/2 if c denotes the Einstein speed of light, so that the two-way speed of light remains equal to c, in agreement with very precise experiments. The present anisotropy refers to a change of formalism, not of physics. It is of some importance that the laws discussed in this paper do not depend on such conventional changes.

Thus: ρ(z) = C exp -w1 z θ , z ≤ h, and ρ(z) = C exp -w2 z+(w1-w2) h θ , z > h, where C is a constant. Integration gives

which coincides with (18). The interest of our method is of course that the Boltzmann factor needs not be postulated.

D General equations of motion

The Hamilton equations of motion in one space dimension (z) derive from a hamiltonian function:

In the situations considered in this paper we may set H(p, z) = F (p) + φ(z), where F (.) is a (nearly) arbitrary function of p, and φ(.) is the potential function. In a static (time-independent) force of constant magnitude w directed along the negative z-axis, we have φ(z) = w z. Thus, a general equation of motion is:

dp , and p = -w (t -t o ), where t o is an integration constant. In particular, for a slow corpuscle of mass m we have: f (p) = p/m. Since w = m g, where g is the acceleration, it follows that z = -1 2 g t 2 , if t o = 0, according to Galileo. For a corpuscle at any speed we have:

where c is the speed of light, according to Einstein. Still other forms, which are dimensionally correct but unlikely to be in general physically significant, would be: f (p) = c g(p/mc) for some function g(.). No particular g(.) function is presumed in this paper.


S Aathithan

G L French

Hypermutability in clinical isolates of Klebsiella pneumoniae is uncommon and is unrelated to ciprofloxacin resistance 

Keywords:

We investigated hypermutability in Klebsiella pneumoniae and its association with ciprofloxacin resistance and mutations in the quinolone resistance-determining region (QRDR). Sixty-four strains of K. pneumoniae isolated in London, UK, between 1995 and 2002 with widely differing ciprofloxacin minimum inhibitory concentrations (MICs) and known gyrA and parC sequences were tested for mutation frequencies by selection with rifampicin. Only three hypermutable (frequency ≥10 -6 ) strains were identified, with ciprofloxacin MICs of 0.25, 8 and 64 g/mL. There was no relationship between hypermutation and the ciprofloxacin MIC or QRDR mutations.

Screening selected strains with streptomycin did not reveal any hypermutators, and screening with ciprofloxacin identified only two of the three hypermutators identified by rifampicin. Hypermutation in K. pneumoniae is uncommon and does not contribute to accumulation of QRDR mutations or directly to ciprofloxacin resistance.

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Introduction

Bacteria generally exhibit a mutation frequency of 10 -10 per replicated base pair [START_REF] Bridges | Hypermutation in bacteria and other cellular systems[END_REF].

Strains that exhibit an elevated mutation frequency are said to be hypermutable [START_REF] Chopra | The role of mutators in the emergence of antibioticresistant bacteria[END_REF][START_REF] Hall | Hypermutable bacteria isolated from humans-a critical analysis[END_REF]. There is no consensus on the mutation frequency above which an isolate is considered to be hypermutable [START_REF] Hall | Hypermutable bacteria isolated from humans-a critical analysis[END_REF], but a frequency of >10 -6 is widely accepted [START_REF] Oliver | High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection[END_REF] and this value has been used in the present study.

The methyl-directed mismatch repair (MMR) system, which involves the genes mutS, mutL, mutH and mutU, normally corrects illegitimate DNA replication [START_REF] Modrich | Methyl-directed DNA mismatch correction[END_REF] and hypermutator phenotypes are thought to arise as a result of mutations in these genes. It is postulated that hypermutators develop antibacterial resistance at a higher frequency than MMR-proficient cells when exposed to antimicrobials [START_REF] Oliver | High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection[END_REF][START_REF] Leclerc | High mutation frequencies among Escherichia coli and Salmonella pathogens[END_REF][START_REF] Blazquez | Hypermutation as a factor contributing to the acquisition of antimicrobial resistance[END_REF]. This may be due to an increased frequency of target site mutations or to changes in other genes, such as those associated with efflux pump activity. Hypermutability may also bring about mutation in a single global gene such as mar, which can lead to the development of simultaneous multiple antibacterial resistance [START_REF] Barbosa | Differential expression of over 60 chromosomal genes in Escherichia coli by constitutive expression of MarA[END_REF]. It has therefore been proposed that hypermutators have a higher rate of antimicrobial resistance and multiple resistance than normal clones and that hypermutation plays a role in the emergence of resistance [START_REF] Chopra | The role of mutators in the emergence of antibioticresistant bacteria[END_REF][START_REF] Oliver | High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection[END_REF][START_REF] Leclerc | High mutation frequencies among Escherichia coli and Salmonella pathogens[END_REF][START_REF] Blazquez | Hypermutation as a factor contributing to the acquisition of antimicrobial resistance[END_REF]. This has been supported by several studies of different bacterial species and antimicrobial agents [START_REF] Oliver | High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection[END_REF][START_REF] Blazquez | Hypermutation as a factor contributing to the acquisition of antimicrobial resistance[END_REF], but not by others [START_REF] Gutierrez | Lack of association between hypermutation and antibiotic resistance development in Pseudomonas aeruginosa isolates from intensive care unit patients[END_REF]. Thus, the general role of hypermutability in the emergence of clinically relevant antibacterial resistance remains uncertain. The incidence of fluoroquinolone resistance in Enterobacteriaceae has steadily increased since these agents were introduced into clinical practice [START_REF] Landman | Evolution of antimicrobial resistance among Pseudomonas aeruginosa, Acinetobacter baumannii and Klebsiella pneumoniae in Brooklyn, NY[END_REF][START_REF] Hooper | Mechanisms of quinolone resistance[END_REF]. The commonest mechanism involved, and the one associated with the highest levels of resistance, is mutations in the quinolone resistance-determining region (QRDR) of the topoisomerase genes, especially gyrA and parC. Accumulation of multiple mutations in this region is associated with increasing fluoroquinolone resistance [START_REF] Hooper | Mechanisms of quinolone resistance[END_REF].

We hypothesised that if hypermutability is associated with the development of fluoroquinolone resistance, then hypermutators should have more QRDR mutations and higher fluoroquinolone minimum inhibitory concentrations (MICs) than nonhypermutators.

We previously determined the QRDR mutations associated with ciprofloxacin resistance in a collection of distinct clinical isolates of Klebsiella pneumoniae with a range of ciprofloxacin MICs [START_REF] Aathithan | Mechanisms and genetics of ciprofloxacin resistance in Klebsiella pneumoniae[END_REF]. In this study, the prevalence of hypermutation in this isolate collection was investigated.

Materials and methods

Organisms, antimicrobials, culture media and antimicrobial susceptibility testing

Sixty-four clinical isolates of K. pneumoniae isolated in London, UK, between 1995 and 2002, chosen to represent distinct isolates by biochemical and antibacterial susceptibility profiles and repetitive sequence-based polymerase chain reaction (repPCR) screening, were investigated [START_REF] Aathithan | Mechanisms and genetics of ciprofloxacin resistance in Klebsiella pneumoniae[END_REF]. Mutations in gyrA and parC were identified by sequencing [START_REF] Aathithan | Mechanisms and genetics of ciprofloxacin resistance in Klebsiella pneumoniae[END_REF]. Ciprofloxacin, rifampicin and streptomycin MICs were

Hypermutability tests 2.2.1. Detection of hypermutability using rifampicin or streptomycin

Hypermutability was investigated by a modified method of Oliver et al. [START_REF] Oliver | High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection[END_REF]. Overnight, aerobic, 37 C shake cultures in Luria-Bertani (LB) broth were used to make cell suspensions of ca. 10 10 colony-forming units (CFU)/mL. The inoculum for the broth culture was taken from an overnight colony on horse blood agar that was streaked to single colonies to check for colony homogeneity. For each isolate, 100 L of cell suspension in phosphate-buffered saline (PBS) were spread on two LB agar plates containing either 100 g/mL rifampicin (ca. 3-6 the rifampicin MICs for the majority of the isolates) or 200 µg/mL streptomycin (two-to several-fold the streptomycin MICs). Plates were incubated at 37 C for 24-48 h and the average number of colonies appearing on the two plates was recorded. Total viable counts were determined on antibacterial-free plates. Each isolate was tested twice and the average mutation frequency was calculated. Putative hypermutable isolates were tested twice more to confirm that this was not a random event. Mutational stability was tested by plating 24 colonies of each isolate on agar containing 100 g/mL rifampicin.

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Page | 7

Detection of hypermutability using ciprofloxacin

A further study was performed to determine hypermutability using ciprofloxacin selection. This study was performed with 12 selected clinical isolates comprising highly resistant, resistant and susceptible phenotypes, with multiple, single and no QRDR mutations, respectively, including the three isolates that were identified as hypermutable by the rifampicin study. Cell suspensions of overnight LB broth cultures of ca. 10 10 CFU/mL were made in PBS and 100 L was spread on two Iso-Sensitest agar (ISA) plates containing ciprofloxacin at 4 the ciprofloxacin MIC and on plates without antimicrobial. Plates were incubated at 37 C and colonies were counted after 24, 48 and 72 h. The frequency of ciprofloxacin-resistant mutant formation was averaged from three experiments. For each isolate, 12 randomly selected colonies from the ciprofloxacin selection plates were tested for stability by subculturing on ISA containing ciprofloxacin at 4 the ciprofloxacin MIC. Additionally, 12 randomly selected colonies of isolates 56 and 38 (hypermutable by the rifampicin study) were subcultured on plates containing 32 g/mL ciprofloxacin to identify highlevel ciprofloxacin-resistant progeny.

Results

Antimicrobial susceptibilities and hypermutability assays

Ciprofloxacin MICs ranged from 0.016 g/mL to >512 g/mL. Rifampicin MICs ranged from 4 g/mL to >256 g/mL, with the majority (75%) being either 16 g/mL or 32 g/mL (ca. 6 or 3 less than the 100 g/mL hypermutability screening concentration). Streptomycin MICs ranged from 0.25 g/mL to >256 g/mL and varied widely. However, 16 of the isolates had similar streptomycin susceptibility and these were used in the streptomycin hypermutability analysis.

Screening for hypermutable isolates using rifampicin and streptomycin, and relationship of hypermutability with ciprofloxacin minimum inhibitory concentration and quinolone resistance-determining region mutations

As shown in Fig. 1, isolates 2, 38 and 56 were hypermutable with respect to rifampicin. None of the 16 selected isolates tested (including 2 that were hypermutable with rifampicin) was hypermutable with respect to streptomycin. Fig. 2 shows that there was no association between ciprofloxacin MICs and rifampicin hypermutability; there was also no association between the number of mutations in the QRDR and rifampicin hypermutability (data not shown). Amongst the three isolates hypermutable with rifampicin, isolate 2 had a ciprofloxacin MIC of 64 g/mL and three QRDR alterations (Ser83Tyr and Asp87Ala in GyrA and Ser80Ile in ParC); isolate 56 had a ciprofloxacin MIC of 8 g/mL with a single Asp87Gly alteration in GyrA; and isolate 38 had a ciprofloxacin MIC of 0.25 g/mL with no QRDR mutations either in gyrA or parC [START_REF] Aathithan | Mechanisms and genetics of ciprofloxacin resistance in Klebsiella pneumoniae[END_REF].

Although some isolates (34, 101, 19, 72, 46 and 102) exhibited elevated mutation frequencies (1.4  10 -7 -4.7 10 -7 ) in the streptomycin assay, none was found to be hypermutable or had an elevated mutation frequency in the rifampicin assay, which is considered to be the 'gold standard'. With the ciprofloxacin assay, isolate 54 showed a near-hypermutant phenotype (7.2  10 -7 ), but all the hypermutant progenies were unstable and it was found to be a non-hypermutator by the other two assays.

Screening for hypermutable isolates using ciprofloxacin, and relationship of hypermutability with ciprofloxacin minimum inhibitory concentration and quinolone resistance-determining region mutations

The ciprofloxacin MICs of the 12 isolates tested for hypermutability by ciprofloxacin screening ranged from 0.032 g/mL to 128 g/mL. They comprised three ciprofloxacin susceptibility groups: six isolates with multiple QRDR mutations that were highly resistant to ciprofloxacin; three isolates with single mutations and lowlevel resistance to ciprofloxacin; and three ciprofloxacin-susceptible isolates with no mutations. The frequency of ciprofloxacin-resistant mutants at 4 the ciprofloxacin MIC varied more than 1000-fold amongst these isolates. Isolates 2 and 38 yielded resistant mutants at frequencies of 4.4  10 -6 and 6.3  10 -6 , respectively, and were thus considered hypermutable. Isolate 56, which had been identified as hypermutable in the rifampicin study, was not identified as hypermutable with ciprofloxacin. The six highly ciprofloxacin-resistant isolates (MICs > 16 g/mL) had multiple mutations in the QRDR, but only isolate 2 was hypermutable by ciprofloxacin screening. Two of the six highly resistant isolates (54 and 45), although not hypermutable, exhibited an elevated mutation frequency of ca. 10 -7 . The three ciprofloxacin-susceptible isolates with MICs of <0.5 g/mL had no mutations in the QRDR, but one (isolate 38) was hypermutable by ciprofloxacin screening and was also hypermutable by the rifampicin assay. Another ciprofloxacin-susceptible isolate (isolate 57) had an elevated mutation frequency of 3.8  10 -7 ; the ciprofloxacin-

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Page | 10 susceptible K. pneumoniae reference strain, with no QRDR mutations, exhibited a similar elevated mutation frequency. In the third group of three isolates with ciprofloxacin MICs of 1-8 g/mL, each had one mutation in the QRDR; none was hypermutable or showed an elevated mutation frequency by ciprofloxacin screening.

None of the 12 colonies of the ciprofloxacin-susceptible but hypermutable isolate 38, randomly selected from the 1 g/mL ciprofloxacin screening plate, grew on 32 g/mL ciprofloxacin. This suggested that these mutant progenies did not inherit multiple QRDR mutations (see discussion), but this was not confirmed by sequencing.

Detection of hypermutability using a modified Etest method

When the Etest was used to determine the MICs of CHL, TET and TMP, a large number of normal-sized squatter colonies was observed within the inhibition zones around the CHL, TET and TMP strips with isolate 38, which was susceptible to these agents and had been identified as hypermutable by rifampicin and ciprofloxacin assays. We concluded that the 'squatter colonies' represented resistant mutant progeny and hypothesised that Etest strips might be used as a simple way to screen for hypermutability with a range of antimicrobials. We therefore tested the presumptive hypermutable isolates 38 and 56 (but not isolate 2, which was highly resistant to all the antibacterials used in the Etest) twice by Etest using ca. 10 7 CFU/89-mm diameter plate (i.e. 10 7 CFU/62 cm 2 ).

As shown in Fig. 3, isolate 38 produced >30 squatter colonies within each inhibition zone, equivalent to ca. 30 colonies/1.25  10 6 CFU seeded, giving a mutation frequency of ca. 2.4  10 -5 with all three antibacterials. This phenomenon was not

Discussion

A collection of 64 distinct K. pneumoniae isolates expressing a range of ciprofloxacin MICs and containing multiple, single or no mutations in the QRDR was investigated.

Hypermutability (mutation frequency of ≥10 -6 ) was uncommon and there was no association between hypermutability and ciprofloxacin resistance or QRDR mutations. Only 3 (<5%) of the 64 isolates were identified as hypermutable by rifampicin screening. Amongst the 24 isolates with multiple QRDR mutations and high-level ciprofloxacin resistance, only one was hypermutable by the rifampicin assay. Furthermore, hypermutability was found more frequently (1 in 3) in isolates with no mutations in the QRDR than in those with multiple mutations (1 in 6). We conclude that there was no causal association between hypermutability and multiple QRDR mutations or high-level ciprofloxacin resistance.

Ciprofloxacin was not used for the initial hypermutability screening because of the possibility of mutants arising as a result of ciprofloxacin SOS induction [START_REF] Phillips | Induction of the SOS response by new 4-quinolones[END_REF][START_REF] Luo | In vivo selection of Campylobacter isolates with high levels of fluoroquinolone resistance associated with gyrA mutations and the function of the CmeABC efflux pump[END_REF]. A relatively high proportion of tested isolates (3/12; 25%) exhibited an elevated frequency of mutation (ca. 10 -7 ) with ciprofloxacin, but this was much lower with rifampicin (2/64; 3.1%). This high prevalence of elevated mutation frequency with ciprofloxacin may be due to the SOS-inducing effect of the drug [START_REF] Phillips | Induction of the SOS response by new 4-quinolones[END_REF][START_REF] Luo | In vivo selection of Campylobacter isolates with high levels of fluoroquinolone resistance associated with gyrA mutations and the function of the CmeABC efflux pump[END_REF]. Isolate 56 appeared hypermutable with rifampicin but not with ciprofloxacin or streptomycin, indicating that apparent hypermutability varies with the agent used for screening. Rifampicin, which generated the least number of isolates with an elevated mutation frequency, appeared to be the best agent for identifying hypermutation in these isolates.

The mutant colonies selected by rifampicin were always stable when subcultured on agar containing 100 g/mL rifampicin. However, many of the apparently resistant colonies that formed on the 4 ciprofloxacin MIC screening plates were unstable when subcultured at the same ciprofloxacin concentration. This striking difference in stability suggests that some or all of the colonies on the 4 ciprofloxacin MIC plates were only transiently resistant or were only 'phenotypic mutators', possibly due to induction of SOS by ciprofloxacin. This indicates that ciprofloxacin is not a reliable agent for identifying hypermutators.

We had previously found that all the isolates in this collection with ciprofloxacin MICs of ≥16 g/mL had multiple QRDR mutations [START_REF] Aathithan | Mechanisms and genetics of ciprofloxacin resistance in Klebsiella pneumoniae[END_REF]. We postulated that progeny of hypermutable isolates selected by ciprofloxacin and resistant to 32 g/mL ciprofloxacin were likely to have multiple QRDR mutations. Colonies of the hypermutable isolate 38 (MIC = 0.25 g/mL) that grew on the 1 g/mL ciprofloxacin hypermutability screening plates were therefore tested for high-level resistance.

None of the 12 randomly selected colonies grew on 32 g/mL ciprofloxacin, suggesting that hypermutability did not give rise to progenies with multiple mutations in the QRDRs in this isolate. We also tested for hypermutation selected by other antimicrobials using Etest strips for 26 of the 64 isolates. This strategy was adopted following the observation that the persistently hypermutant isolate 38 consistently yielded a large number of normalsized squatter colonies within the Etest zones of inhibition for CHL, TET and TMP (Fig. 3). Amongst the three hypermutable isolates identified by rifampicin screening, the Etest method showed that only isolate 38 consistently expressed the hypermutable phenotype with CHL, TET and TMP. Thus, only isolate 38 was consistently hypermutable with a range of antimicrobials. We propose that hypermutability should be tested by selection with a range of antimicrobial classes and that this can be done conveniently by Etest. Thus, our study lends further support for the proposal that the Etest, which is simpler than the commonly used hypermutation assay, could be used to identify hypermutators in clinical laboratories [START_REF] Maciá | Detection and susceptibility testing of hypermutable Pseudomonas aeruginosa strains with the Etest and disk diffusion[END_REF]. Looking for hypermutation in routine Etest-based MIC plates could help alert clinicians about the possibility of enhanced resistance development in isolates that produce normal-sized resistant colonies in the zone of inhibition.

In conclusion, this study found that: (i) hypermutability was uncommon in clinical isolates of K. pneumoniae in London; (ii) hypermutability occurred randomly amongst ciprofloxacin-susceptible and -resistant strains and there was no evidence to support the hypothesis that hypermutability was associated with or contributed to the development of multiple mutations in the QRDR or resistance to ciprofloxacin; (iii) the progeny of hypermutable isolates did not show high-level ciprofloxacin resistance, suggesting that hypermutability did not produce multiple mutations in the QRDR; (iv) the apparent hypermutability of an isolate depends on the agent used in Isolates are plotted on the x-axis and mutation frequencies are on the y-axis. Mutator strains are defined as having a mutation frequency of ≥10 -6 . Note that the isolate number on the x-axis does not correspond to the true isolate numbers, referred to in the text.

2 

 2 Part of this study was presented at the 19th European Congress of Clinical Microbiology and Infectious Diseases (ECCMID), 16-19 May 2009, Helsinki, Finland (abstract P1481).

  dilution. MICs of chloramphenicol (CHL), tetracycline (TET) and trimethoprim (TMP) were determined for 26 isolates by Etest. Escherichia coli NCTC 10418 and K. pneumoniae NCTC 009633 were used as standards and as controls in the hypermutability studies.

  A c c e p t e d M a n u s c r i p t Page | 8

A

  c c e p t e d M a n u s c r i p t Page | 11 seen with isolate 56, another presumptive hypermutable isolate selected with rifampicin, or with other non-hypermutable isolates.

Fig. 1 .

 1 Fig.1. Frequency of mutation with rifampicin in 64 Klebsiella pneumoniae isolates.

Fig. 2 .

 2 Fig. 2. Ciprofloxacin (CIP) minimum inhibitory concentrations (MICs) and frequency

Fig. 3 .

 3 Fig. 3. Hypermutable isolate 38 showing hypermutability using Etest strips. Strips

Acknowledgment

The authors would like to thank Dr K. Shannon for advice and help with computer analysis of the data.

Funding

SA and the study were funded by King's College London (UK).

A c c e p t e d M a n u s c r i p t Page | 14 screening; (v) the use of ciprofloxacin in hypermutability screening studies may result in an apparent elevated mutation frequency, possibly due to SOS response of nonheritable nature; and (vi) putative hypermutant progeny should always be tested for stability before concluding that the parent strain is a true hypermutator.

Competing interests

None declared.

Ethical approval

Not required.


I Galani
email: egalani@med.uoa.gr

M Souli

N Mitchell

Z Chryssouli

H Giamarellou

Presence of plasmid-mediated quinolone resistance in Klebsiella pneumoniae and Escherichia coli isolates possessing bla VIM-1 in Greece

Keywords:

Among nalidixic acid-resistant, ciprofloxacin-susceptible Escherichia coli and Klebsiella pneumoniae clinical isolates recovered over a 5-month period from inpatients and outpatients of Attikon University General Hospital (Athens, Greece), only one E. coli was positive for qnrB2 and one K. pneumoniae was positive for qnrA1. Both isolates were extended-spectrum -lactamase-negative, metallo-lactamase-positive and carried the bla VIM-1 gene. Neither of the isolates had mutations in gyrA and parC or carried aac(6')-Ib-cr or qepA. The K. pneumoniae isolate also harboured bla CMY-13 on the same transferable plasmid with qnrA1. This is the first report of a qnrA1-positive K. pneumoniae and qnrB2-positive E. coli harbouring a concurrent bla VIM-1 gene.

Introduction

Quinolone resistance in Enterobacteriaceae is mostly mediated by point mutations in the quinolone resistance-determining regions (QRDRs) of the gyrase and topoisomerase IV genes, leading to target modification [START_REF] Robicsek | The worldwide emergence of plasmidmediated quinolone resistance[END_REF]. More recently, three plasmid-mediated quinolone resistance mechanisms have been described: Qnr peptides, capable of protecting DNA gyrase and topoisomerase IV from quinolones; Aac(6')-Ib-cr aminoglycoside acetyltransferase, modifying quinolones with a piperazinyl substituent (e.g. ciprofloxacin); and the quinolone efflux pump QepA [START_REF] Robicsek | The worldwide emergence of plasmidmediated quinolone resistance[END_REF].

qnrA1 from a clinical strain of Klebsiella pneumoniae isolated in Alabama was first described by Martinez-Martinez et al. in 1998 [START_REF] Martinez-Martinez | Quinolone resistance from a transferable plasmid[END_REF]. To date, qnrA, qnrB and qnrS comprise three families differing from each other by 40% in nucleotide sequence.

Within each family, minor (≤10%) variation in sequence has defined a growing number of alleles. A total of 6 qnrA, 23 qnrB, 4 qnrS and 1 each of qnrC and qnrD variants have been described in the literature and are listed in the database maintained at the website http://www.lahey.org/qnrStudies (updated 16 February 2010).

qnr genes are widely distributed in clinical Enterobacteriaceae isolates around the world and are usually associated with mobile genetic elements (complex sul1-type integrons) [START_REF] Robicsek | The worldwide emergence of plasmidmediated quinolone resistance[END_REF]. Their products lead to increased minimum inhibitory concentrations (MICs) of quinolones, which in many cases are still below the approved susceptibility breakpoints and require the coexistence of an additional mechanism to lead to frank resistance [START_REF] Robicsek | The worldwide emergence of plasmidmediated quinolone resistance[END_REF]. qnr genes were found to be carried on the same plasmid with various extended-spectrum or AmpC-type -lactamases genes [START_REF] Robicsek | The worldwide emergence of plasmidmediated quinolone resistance[END_REF]. Recently, Mendes et al. [START_REF] Mendes | Carbapenemresistant isolates of Klebsiella pneumoniae in China and detection of a conjugative plasmid (bla KPC-2 plus qnrB4) and a bla IMP-4 gene[END_REF] A c c e p t e d M a n u s c r i p t and Endimiani et al. [START_REF] Endimiani | Presence of plasmid-mediated quinolone resistance in Klebsiella pneumoniae isolates possessing bla KPC in the United States[END_REF] reported the presence of plasmid-mediated quinolone resistance in K. pneumoniae isolates possessing bla KPC in China and the USA, respectively, whilst Chmelnitsky et al. [START_REF] Chmelnitsky | Plasmid-mediated qnrB2 and carbapenemase gene bla KPC-2 carried on the same plasmid in carbapenem-resistant ciprofloxacin-susceptible Enterobacter cloacae isolates[END_REF] reported the presence of bla KPC-2 and qnrB2 on the same plasmid in Enterobacter cloacae isolates. The metallo--lactamase (MBL) IMP-8 has been associated with qnrB and qnrS in E. cloacae isolates in Taiwan, whilst bla VIM-1 was located on the same plasmids as qnrS1 in multiple species of Enterobacteriaceae in Italy [START_REF] Wu | Prevalence of plasmid-mediated quinolone resistance determinants QnrA, QnrB, and QnrS among clinical isolates of Enterobacter cloacae in a Taiwanese hospital[END_REF][START_REF] Aschbacher | Linkage of acquired quinolone resistance (qnrS1) and metallo--lactamase (bla VIM-1 ) genes in multiple species of Enterobacteriaceae from Bolzano, Italy[END_REF]. Very recently Miró et al. [START_REF] Miró | Spread of plasmids containing the bla VIM-1 and bla CTX-M genes and the qnr determinant in Enterobacter cloacae, Klebsiella pneumoniae and Klebsiella oxytoca isolates[END_REF] showed a linkage between bla VIM-1 , qnrS and aac(6')-Ib-cr genes and between bla CTX-M-9 and qnrA genes identified in Klebsiella and Enterobacter spp.

We report the isolation of a K. pneumoniae and an Escherichia coli isolate, both ciprofloxacin-susceptible but nalidixic acid-resistant, possessing bla VIM-1 on the same conjugative plasmid as qnrA and qnrB, respectively.

Material and methods

The computerised database of the hospital's Infectious Diseases Research Laboratory was retrospectively searched for E. coli and K. pneumoniae strains that were nalidixic acid-resistant and ciprofloxacin-susceptible isolated during a 5-month period (February-June 2008) from inpatients and outpatients of Attikon University General Hospital (Athens, Greece). Only one isolate per patient was included in the study. Species identification of isolated bacteria and MIC determinations were performed using an automated system (BD Phoenix automated microbiology system; BD Diagnostic Systems, Sparks, MD). MICs of nalidixic acid were also evaluated by broth microdilution in accordance with Clinical and Laboratory Standards Institute (CLSI) guidelines [START_REF]Performance standards for antimicrobial susceptibility testing. Nineteenth informational supplement[END_REF], whereas MICs of ciprofloxacin, moxifloxacin, imipenem, A c c e p t e d M a n u s c r i p t meropenem, doripenem, aztreonam and trimethoprim/sulfamethoxazole (SXT) were determined by Etest (AB BIODISK, Solna, Sweden) in accordance with the manufacturer's instructions. Results were interpreted applying CLSI criteria for in vitro susceptibility [START_REF]Performance standards for antimicrobial susceptibility testing. Nineteenth informational supplement[END_REF]. All isolates were screened for extended-spectrum -lactamase (ESBL) production by the double-disk synergy test and the CLSI confirmatory test, for MBL production using the ethylene diamine tetra-acetic acid (EDTA)-imipenem disk synergy test, and for KPC production by the imipenem-boronic acid disk synergy test [START_REF]Performance standards for antimicrobial susceptibility testing. Nineteenth informational supplement[END_REF][START_REF] Galani | Evaluation of different laboratory tests for the detection of metallo--lactamase production in Enterobacteriaceae[END_REF][START_REF] Tsakris | Evaluation of boronic acid disk tests for differentiating KPC-possessing Klebsiella pneumoniae in the clinical laboratory[END_REF].

Isolates were screened for the presence of qnrA, qnrB and qnrS by multiplex polymerase chain reaction (PCR) using universal primers for each gene amplifying all related alleles [START_REF] Cattoir | Multiplex PCR for detection of plasmid-mediated quinolone resistance qnr genes in ESBL-producing enterobacterial isolates[END_REF]. Conjugation experiments were carried out by the filter method with the two qnr-positive clinical isolates as donors and a rifampicin-resistant E. coli K-12 RC85 laboratory strain as a recipient. Transconjugant clones were selected on brainheart infusion agar (Bio-Rad, Marnes-la-Coquette, France) plates containing 150 g/mL rifampicin and 8 g/mL nalidixic acid. qnr-positive isolates and their transconjugants were additionally tested by isoelectric focusing of sonic extracts on pre-cast polyacrylamide gels with a pH 3-10 gradient (PhastGel IEF 3-9; Amersham Biosciences, Freiburg, Germany) for -lactamase detection. The presence of bla VIM, bla CMY, bla SHV and bla TEM was confirmed by PCR with specific primers shown in Table 1. The genes aac(6')-Ib and qepA were detected by PCR with specific primers (Table 1

Results and discussion

Among 231 single-patient E. coli and 231 single-patient K. pneumoniae isolates found on the database during the study period, 13 and 11 isolates, respectively, were resistant to nalidixic acid and susceptible to ciprofloxacin and were studied further.

Ciprofloxacin MIC values ranged from 0.125 g/mL to 1 g/mL for E. coli and 0.125 g/mL to 0.5 g/mL for K. pneumoniae. Escherichia coli m4353, which was isolated from a faecal specimen of a patient hospitalised in the Intensive Care Unit of Attikon Hospital, was positive for qnrB. Klebsiella pneumoniae U8699, which was isolated from a urine specimen of a patient with a history of multiple hospitalisations in Attikon Hospital, was positive for qnrA.

Sequencing revealed the existence of qnrB2 and qnrA1 alleles, respectively. No isolates from those collected were positive for qnrS. Both qnrA1-and qnrB2-positive isolates were ESBL-negative and MBL-positive by phenotypic tests and harboured the bla VIM-1 gene. Klebsiella pneumoniae U8699 in addition to VIM-1 carbapenemase coded for CMY-13 [isoelectric point (pI) 9.1] and the intrinsic SHV-11 (pI 7.6), whilst E. coli m4353 coded for VIM-1 and a TEM-1-like enzyme. aac(6')-Ib-cr variant and qepA were not detected in either isolate and no mutations were detected in gyrA and parC.

A c c e p t e d M a n u s c r i p t

Results of conjugation experiments showed that qnrA1 and qnrB2 were located on conjugative plasmids of K. pneumoniae and E. coli, respectively. The plasmid containing qnrA1 also contained bla VIM-1 and bla CMY-13 and was transferred to the E. coli recipient at a frequency of 10 -5 transconjugants per recipient, resistance to nalidixic acid, several -lactams, inhibitor combinations and SXT (Table 2). The qnrA1 transconjugant exhibited decreased susceptibility to carbapenems (imipenem MIC = 1 g/mL; meropenem MIC = 0.19 g/mL) and aztreonam (MIC = 0.2 g/mL).

The plasmid carrying qnrB2 also contained bla VIM-1 and bla TEM-1-like and was transferred to the E. coli recipient at a frequency of 10 -4 transconjugants per recipient, conferring resistance to nalidixic acid, several -lactams, inhibitor combinations, tetracycline and SXT (Table 2). The qnrB2 transconjugant exhibited decreased susceptibility to carbapenems (imipenem MIC = 4 g/mL; meropenem MIC = 0.19 g/mL) and amikacin (MIC = 16 g/mL).

Vasilaki et al. [START_REF] Vasilaki | Emergence of the plasmid-mediated quinolone resistance gene qnrS1 in Escherichia coli isolates in Greece[END_REF] have reported that a considerably high proportion (10%) of quinolone-resistant E. coli isolates from northern and central Greece were found to carry qnrS1 in addition to mutations in gyrA (S83L, D87N) and parC (mainly S80I).

None of these isolates carried qnrA or qnrB, whilst only one was an ESBL-producer, carrying bla CTX-M-15 . Our findings indicate that qnrA1 and qnrB2 are also present in E. coli and K. pneumoniae clinical isolates in Greece and they are not always associated with the QRDR mutations.

A c c e p t e d M a n u s c r i p t

The gene bla CMY-13 , which codes for a cephalosporinase with activity similar to that of CMY-2, has been previously reported to be located on an IncN plasmid from an coli isolate from Greece [START_REF] Miriagou | CMY-13, a novel inducible cephalosporinase encoded by an Escherichia coli plasmid[END_REF]. This plasmid carried bla VIM-1 , aacA7, dhfrI and aadA gene cassettes on a class I integron, but not a qnr determinant [15].

In conclusion, this is the first report of the presence of qnrA1 and of qnrB2 on the same transferable plasmid with bla VIM-1 and bla CMY-13 in K. pneumoniae and with bla VIM-1 in E. coli, respectively. Although the qnr genes are responsible for low-level resistance to quinolones, they are easily transferable and they provide a favourable background for higher level resistance to occur if other mechanisms coexist, such as secondary changes in DNA gyrase or topoisomerase IV, porin or efflux systems. The presence of mechanisms for broad-spectrum resistance, such as genes coding for carbapenemases on the same plasmid, highlights the clinical importance of these genes and the potential for selection and dissemination of resistance to various antimicrobials through improper quinolone use.

A

  c c e p t e d M a n u s c r i p t Sequencing of PCR products was performed by Eurofins MWG Operon (Ebersberg, Germany). For sequence analysis, the BLAST program from the National Center for Biotechnology Information website was used (http://www.ncbi.nlm.nih.gov/BLAST).

  s c r i p t [15] Miriagou V, Tzelepi E, Gianneli D, Tzouvelekis LS. Escherichia coli with a selftransferable, multi-resistant plasmid coding for the metallo--lactamase VIM-1. Antimicrob Agents Chemother 2003;47:395-7.

  ). QRDRs of gyrA and parC were amplified with primers shown in Table 1 and were sequenced to evaluate possible coexisting chromosomal mutations.

Table 1

 1 Primers used in this study

	Primer a Sequence (5'3')	Gene	Size of PCR-amplified
				product (bp)
	AAC-cr-	TTGCGATGCTCTATGAGTGGCTA aac(6')-	482
	F		Ib
	AAC-cr-	CTCGAATGCCTGGCGTGTTT
	R
	QEPA-F CGTGTTGCTGGAGTTCTTC	qepA	403
	QEPA-R CTGCAGGTACTGCGTCATG
	VIM-F	AGTGGTGAGTATCCGACAG	bla VIM	509
	VIM-R	ATGAAAGTGCGTGGAGAC
	SHV-F	ATGCGTTAT ATTCGCCTGTG	bla SHV	865
	SHV-R	GTTAGCGTTGCCAGTGCTCG
	CMY-F	CTGCTGCTGACAGCCTCTTT	bla CMY	1108
	CMY-R TTTTCAAGAATGCGCCAGGC
	TEM-F	CATTCAAATATGTATCCGCTCATG bla TEM	745
	TEM-R	CATCCATAGTTGCCTGACTCCCC
	GYR-A-	TTAATGATTGCCGCCGTCGG	gyrA	648
	F
	GYR-A-	TACACCGGTCAACATTGAGG
	R
	PAR-C-	AAACCTGTTCAGCGCCGCATT	parC	395
	F
	PAR-C-	GTGGTGCCGTTAAGCAAA
	R
	PCR, polymerase chain reaction.
	a F, sense primer; R, antisense primer.

Table 2

 2 Minimum inhibitory concentrations (MICs) of clinical and laboratory isolates used in the study

	Antimicrobial agent	MIC (mg/L)
		Escherichia coli K-	Klebsiella	E. coli U8699	E. coli	E. coli m4353
		12 RC85 recipient	pneumoniae	transconjugant	m4353	transconjugant
			U8699
	Amikacin	8	16	8	32	16
	Amoxicillin/clavulanic acid	4/2	>16/8	>16/8	>16/8	>16/8
	Ampicillin	4	>16	>32	>16	>32
	Aztreonam a	0.032	4	0.2	0.094	0.032
	Cefepime	2	>16	>16	>16	>16
	Cefotaxime	1	>32	32	>32	>32
	Cefoxitin	4	>16	>16	>16	8
	Ceftazidime	1	>16	>16	>16	>16
	Cefuroxime	4	>16	>16	>16	>16
	Cefalothin	8	>16	>16	>16	>16
	Ciprofloxacin a	0.16	0.38	0,19	0,25	0,19
	Colistin	0.5	N/D	0.5	N/D	0.5
	Doripenem a	0.23	6	0.25	0.19	0.38
	Gentamicin	2	8	2	4	2

N/D, not determined; R, resistant; PCR, polymerase chain reaction. a MICs determined by Etest. b MICs evaluated by broth microdilution. c Susceptibility determined by disk diffusion test.

Acknowledgments

The authors would like to thank K. Orlandou for technical assistance.

Funding

None.

Competing interests

None declared.