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Supplementary materials
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The method descrided here are described in detail by Li et al. (2015), for more
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detailed information, please refer to their publication.
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Estimation of evaporation rate and effective diffusion coefficient in biofilm
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The porosity of the biofilm (θ) was estimated from the volume of cells (biovolume) in
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the biofilm divided by the total biofilm volume. The biovolume was calculated as
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described by Hillebrand et al. (1999) from suspended biofilm samples with a 50 μm
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resolution by means of a custom-made hand microtome (Fig. S2). The biofilm
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immobilized on a polycarbonate membrane was placed onto a height-adjustable stage
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on a thin layer of culture medium. The stage allows levelling the biofilm sample into
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the sectioning plane by means of a micrometer screw. The sectioning plane was
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defined by a solid brass support over which a rigid microtome blade was passed
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manually to remove the overlying algal biofilm (Fig. S2).
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Evaporation rate during the measurement was 0.68 mL min-1 (monitored by recording
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the change of the volume in the medium container, Fig S1), which corresponded to a
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convective flow rate of 5.4 µm s-1 inside the biofilm, which was calculated as:
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𝑢 = 𝑞𝑒 /(𝐴𝑒 ∙ 𝜃)
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(Eq.S1),
qe is the rate of liquid volume lost, Ae is the exposed surface area of the biofilm.
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Microsensor setup
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A schematic representation of the experimental setup for microsensor measurements
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based on previous works by Gieseke and de Beer (2004) is given in Fig. S1: the
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biofilm immobilized on a polycarbonate membrane was placed onto a moist glass
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fiber inside the biofilm measurement cell under a controlled atmosphere of
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compressed air at a flow rate of 1 L min-1 (Fig. S1A). The non-inoculated area of the
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polycarbonate filter and glass fiber were covered with a black plastic foil to exclude
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light and gas exchange in this area. 50 mL of BBM culture medium were circulated
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though the measurement cell with a flow rate of 3 mL min-1 by means of a peristaltic
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pump and were exchanged every 1 hour during the measurement. Light was supplied
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from the front side by a halogen lamp (KL-1500, Schott, Mainz, Germany) equipped
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with a 3-fold splitter to ensure even distribution. The movement of the sensor was
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enabled by a computer-controlled micromanipulator (Pollux Drive, PI miCos,
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Eschbach, Germany). Microsensor signals were amplified and converted into digital
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data (DAQpad 6015 and 6009, National Instruments, Munich, Germany) prior to their
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further processing on a computer (Fig. S1B). Software used for system control and
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data acquisition was custom made (can be acquired upon request from Dr. Lubos
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Polerecky, Utrecht University, Netherlands, l.polerecky@uu.nl). Light-dark shift
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measurements (Revsbech, 1989) were carried out, and the data acquired was
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processed as described in the following section. Linear calibration of oxygen
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microsensor was carried out as described by Revsbech (1983) using BBM medium
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saturated with nitrogen gas or compressed air. A shutter connected to a timer was
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used to control light-dark cycling: the total dark period was 3 s, and a linear
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regression slope of oxygen concentration change between 0.7 and 2.1 s of the dark
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period was determined as the measured value. For each measurement depth, three
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light-dark cycles were performed after steady state was reached (this could be as long
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as 10 min), and these 3 values acquired were considered as triplicates. The PAR light
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intensity of a fixed depth was determined as the average value of a 3 s measurement
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period.
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Gross photosynthetic productivity calculation
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Take the derivative of time on both sides of the dissolved oxygen mass transfer
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equation, and assume the time t and spatial variables x are not dependent (as during
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the measurement period, the thickness of the biofilm hardly changes) (Bergman et al.,
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2011, Revsbech and Jorgensen, 1983):
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𝜕𝐶
𝜕𝐶
𝜕2 ( )
𝜕( )
𝜕𝐶
𝜕𝑡 + 𝑢
𝜕𝑡 − 𝜕𝑅 /𝜕𝑡
𝜕( )/𝜕𝑡 = 0/𝜕𝑡 − 𝜕𝑅𝑙 /𝜕𝑡 + 𝐷𝑒
𝑠
𝜕𝑡
𝜕𝑥 2
𝜕𝑥
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(Eq.S2).
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The term ∂Rs/∂t represents the rate change of the removal of dissolved gaseous
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species at the submerged biofilm surface due to the flow of bulk liquid, or, in this
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case, of the gas phase above the biofilm:
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𝜕𝑅𝑠 /𝜕𝑡 = (𝐷𝑒
(𝐶 − 𝐶𝑠 )
(𝐶 − 𝐶𝑠 )
+
𝑢
)/𝜕𝑡
𝑥2
𝑥
3
=(
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𝐷𝑒 𝑢
+ ) ∙ 𝜕𝐶/𝜕𝑡
𝑥2 𝑥
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(Eq.S3).
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Cs represents the oxygen concentration on the very surface of the aerial biofilm,
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which is in equilibrium with the phase above and is considered to be a constant.
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Considering that the rate of respiration remains stable during the measurement period
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(Glud et al., 1992), and take ∂C/∂t=P(x, t), as ∂C/∂t is a function of depth x and time t.
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Substitute Eq.S3 into ∂Rs/∂t and ∂C/∂t to P(x, t), Eq.S2 becomes:
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𝜕 2 (𝑃(𝑥, 𝑡))
𝜕(𝑃(𝑥, 𝑡))
𝜕𝑃(𝑥, 𝑡)
𝐷𝑒 𝑢
= 0 − 0 + 𝐷𝑒
+𝑢
− ( 2 + ) ∙ 𝑃(𝑥, 𝑡)
2
𝜕𝑡
𝜕𝑥
𝜕𝑥
𝑥
𝑥
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(Eq.S4).
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Eq.S4 is a nonhomogeneous diffusion equation with a sink and can be solved by using
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any of the methods summarized by Crank or Polyanin (Crank, 1975; Polyanin, 2002).
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Using an initial condition P(x, t = 0) = P(x, 0) and a boundary condition ∂P(x = 0,
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t)/∂t = 0, the exact solution of Eq.S4 at t = τ can be expressed as:
𝛿
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𝑃(𝑥, 𝜏) = ∫ 𝑃(𝑦, 0) ∙ [𝑒
−(𝑦−𝑥+𝑢∙𝜏)
(
)
4∙𝐷𝑒 ∙𝜏
0
−𝑒
−(𝑦+𝑢∙𝜏)
(
)
4∙𝐷𝑒 ∙𝜏
∙ 𝑒𝑟𝑓𝑐(
𝑥
√4 ∙ 𝐷𝑒 ∙ 𝜏
)] ∙ 𝑑𝑦
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(Eq.S5),
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δ is the total depth of the biofilm, or, alternatively, the depth until which the shift from
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light to dark has an effect on the dissolved oxygen concentration. P(x, τ) is the
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measured value from the light-dark shift method with measurement time τ at
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measurement depth x, and P(y, 0) is the actual photosynthetic activity at position y
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(the integrating variable), which is the desired term. For measurements taken at all
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depths, the measured value P(x, τ) at a depth x depends on the P(y, 0) across the
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complete photosynthetically active region of the biofilm (surface to depth δ).
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Eq.S5 is a Fredholm integral equation of the first kind (Kress, 2014), and its
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approximated solution can be calculated using Tikhonov regularization (Tikhonov et
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al., 1995) with a non-negative constrain applying the active-set algorithm purposed by
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Gill et al. (1981): Discrete δ by n, and let P(xj, τ) = gj, P(yi, 0) = hi; (i, j = 1, 2, …, n)
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and write Eq.S5 in the discrete operator notion.
𝑛
𝑔𝑗 = ∑ 𝑘𝑖,𝑗 ∙ ℎ𝑖
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𝑖
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(Eq.S6),
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ki,j is the operator term which contains both the term that reflects the effect of
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diffusion and advection of dissolved oxygen inside the biofilm (inner term, first term
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in square bracket in Eq.S5) and the term that reflects the effect of dissolved oxygen
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removal due to surface flow (surface term, second term in square bracket in Eq.S5). In
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matrix notion, let G be a vector containing gj, H a vector containing hi, and K a matrix
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containing the operator terms ki,j:
𝐺 =𝐾∙𝐻
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(Eq. S7).
Rewrite Eq.S7 as:
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(𝐺𝑖𝑛 + 𝐺𝑠𝑢𝑟𝑓 ) = (𝐾𝑖𝑛 + 𝐾𝑠𝑢𝑟𝑓 ) ∙ 𝐻
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(Eq.S9),
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The subscripts in and surf represent the effect from the inner term and the surface
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term in Eq.S5 respectively, operators Kin, Ksurf can be calculated using Eq.S5, as well
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as gin and Gsurf, if an H is given. Apply the Tikhonov regularization, the problem in
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Eq.11 thus becomes:
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min {‖(𝐺𝑖𝑛 + 𝐺𝑠𝑢𝑟𝑓 ) − (𝐾𝑖𝑛 + 𝐾𝑠𝑢𝑟𝑓 ) ∙ 𝐻‖ + 𝜆2 ∙ ‖𝐿 ∙ 𝐻‖2 }
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𝐹≥0
or,
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min {‖(𝐺𝑖𝑛 − 𝐾𝑖𝑛 ∙ 𝐻) + (𝐺𝑠𝑢𝑟𝑓 − 𝐾𝑠𝑢𝑟𝑓 ∙ 𝐻)‖ + 𝜆2 ∙ ‖𝐿 ∙ 𝐻‖2 }
𝐹≥0
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(Eq.S10).
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Observing the surface term in Eq.S5, notice the maximum value of the surface term is
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controlled by the position of the actual activity y, but a complement error function
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dependent solely on the position of the measurement taken (x) controls the final value.
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Furthermore, for a given G, under the same measurement conditions, Gsurf always has
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the same set of values. Thus, for a given G, the minimization of the term
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‖𝐺𝑠𝑢𝑟𝑓 − 𝐾𝑠𝑢𝑟𝑓 ∙ 𝐻‖ will always yield the same solution. As a result, the problem in
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Eq. S10 can be simplified to:
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2
min{‖𝐺𝑖𝑛 − 𝐾𝑖𝑛 ∙ 𝐻‖2 + 𝜆2 ∙ ‖𝐿 ∙ 𝐻‖2 }
𝑓≥0
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(Eq. S11).
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Eq.S11 does not contain the surface term, but the solution of the problem still leads to
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H, which is the desired original profile. The L-curve method purposed by Hansen &
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O'Leary (1993) was used to find the best regularization parameter λ.
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The treatment procedure was coded in MATLAB (version 2013a, MathWorks,
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Ismaning, Germany), and the inbuilt quadprog was used for solving the minimization
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problems. L-curve function from the regularization tools (Hansen, 1994) was used to
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find the λ value. The mean values of triplicates acquired from the microsensor
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measurement were randomly added or subtracted with a random value in range of the
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calculated standard deviation (SD) at the same position. These values were used as
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input for the calculation. The calculation procedure was repeated 3 times, and the
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mean value of the 3 results was taken as the final result (a SD can also be calculated).
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Data interpolation, if needed, was done by simply connecting two measured data
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points with a straight line (linear interpolation, using MATLAB inbuilt function
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interp1).
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References for supplementary materials:
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Bergman TL, Lavine AS, Incropera FP, DeWitt DP. 2011. Fundamentals of heat and
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mass transfer. 7th ed. Hoboken, New Jersey, USA: John Wiley & Sons. 1048 p.
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Crank J, editor. 1975. Mathematics of diffusion. 2nd ed. Oxford, UK: Oxford
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University Press. 414 p.
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Gieseke A, de Beer D. 2004. Use of microelectrodes to measure in situ microbial
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activities in biofilms, sediments, and microbial mats. In: Kowalchuk GA, de Bruijn
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FJ, Head IM, Akkermans AD, van Elsas JD, editors. Molecular Microbial Ecology
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Manual. The Netherlands: Springer. p 3483-3514.
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Gill PE, Murray W, Wright MH. 1981. Practical optimization. Waltham,
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Massachusetts, USA: Academic Press. 401 p.
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Glud RN, Ramsing NB, Revsbech NP. 1992. Photosynthesis and photosynthesis-
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coupled respiration in natural biofilms quantified with oxygen microsensors. Journal
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of Phycology 28(1):51-60.
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Hansen P. 1994. REGULARIZATION TOOLS: A Matlab package for analysis and
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solution of discrete ill-posed problems. Numerical Algorithms 6(1):1-35.
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Hansen PC, O’Leary DP. 1993. The Use of the L-Curve in the Regularization of
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Discrete Ill-Posed Problems. SIAM Journal on Scientific Computing 14(6):1487-
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1503.
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Hillebrand H, Durselen CD, Kirschtel D, Pollingher U, Zohary T. 1999. Biovolume
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calculation for pelagic and benthic microalgae. Journal of Phycology 35(2):403-424.
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Kress R, editor. 2014. Linear integral equations. 3rd ed. New York City, USA:
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Springer. 412 p.
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Li T, Podola B, de Beer D, Melkonian M. 2015. A method to determine
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photosynthetic activity from oxygen microsensor data in biofilms subjected to
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evaporation. Journal of Microbiological Methods 117:100-107.
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Polyanin AD. 2002. Handbook of linear partial differential equations for engineers
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and scientists. Boca Raton, Florida: Chapman&Hall/CRC Press. 800 p.
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Revsbech NP. 1983. In Situ Measurement of Oxygen Profiles of Sediments by use of
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Oxygen Microelectrodes. In: Gnaiger E, Forstner H, editors. Polarographic Oxygen
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Sensors. Berlin/Heidelberg, Germany: Springer. p 265-273.
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Revsbech NP, Jorgensen BB. 1983. Photosynthesis of benthic microflora measured
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with high spatial-resolution by the oxygen microprofile method - capabilities and
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limitations of the method. Limnology and Oceanography 28(4):749-756.
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Tikhonov AN, Goncharsky A, Stepanov VV, Yagola AG. 1995. Numerical methods
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for the solution of ill-posed problems. Dordrecht, the Netherlands: Springer. 254 p.
Supplementary materials figure captions
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Figure S1
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Experimental setup for microsensor studies in non-submerged microalgal biofilms. A:
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Schematic drawing of the vertical cross section of the measurement in artificial
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biofilms. Solid arrows in the glass fiber tissue indicate flow direction of the culture
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medium. B: Schematic drawing of the complete experimental setup, dotted lines with
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arrow indicate signal flow, whereas solid arrows represent medium and gas flows in
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the system.
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Figure S2
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Schematic drawing of the experimental setup used for biofilm sectioning. The sample
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holder allows moving the biofilm in vertical direction by means of a micrometer
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screw (indicated by the thin solid arrows). The biomass above the sectioning plane
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was removed with a solid microtome blade. The cutting direction is indicated by the
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thick arrow.
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